author | huffman |
Sun, 13 May 2007 20:05:42 +0200 | |
changeset 22956 | 617140080e6a |
parent 22943 | 0b928312ab94 |
child 22961 | e499ded5d0fc |
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 {* Nth Roots of Real Numbers *} |
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theory NthRoot |
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imports SEQ Parity |
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begin |
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subsection {* Existence of Nth Root *} |
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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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subsubsection {* 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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subsubsection {* 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_of_nat_inverse_le_iff: |
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"(inverse (real(Suc n)) \<le> r) = (1 \<le> r * real(Suc n))" |
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by (simp add: inverse_eq_divide pos_divide_le_eq) |
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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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text {* 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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text {* 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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text {* We define roots of negative reals such that |
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@{term "root n (- x) = - root n x"}. This allows |
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us to omit side conditions from many theorems. *} |
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definition |
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root :: "[nat, real] \<Rightarrow> real" where |
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"root n x = (if 0 < x then (THE u. 0 < u \<and> u ^ n = x) else |
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if x < 0 then - (THE u. 0 < u \<and> u ^ n = - x) else 0)" |
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lemma real_root_zero [simp]: "root n 0 = 0" |
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unfolding root_def by simp |
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lemma real_root_minus: "0 < n \<Longrightarrow> root n (- x) = - root n x" |
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unfolding root_def by simp |
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lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 0 < root n x" |
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apply (simp add: root_def) |
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apply (drule (1) realpow_pos_nth_unique) |
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apply (erule theI' [THEN conjunct1]) |
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done |
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lemma real_root_pow_pos: (* TODO: rename *) |
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"\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> root n x ^ n = x" |
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apply (simp add: root_def) |
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apply (drule (1) realpow_pos_nth_unique) |
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apply (erule theI' [THEN conjunct2]) |
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done |
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lemma real_root_pow_pos2 [simp]: (* TODO: rename *) |
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"\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x" |
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by (auto simp add: order_le_less real_root_pow_pos) |
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|
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lemma real_root_ge_zero: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> 0 \<le> root n x" |
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by (auto simp add: order_le_less real_root_gt_zero) |
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|
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lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x" |
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apply (subgoal_tac "0 \<le> x ^ n") |
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apply (subgoal_tac "0 \<le> root n (x ^ n)") |
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apply (subgoal_tac "root n (x ^ n) ^ n = x ^ n") |
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apply (erule (3) power_eq_imp_eq_base) |
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apply (erule (1) real_root_pow_pos2) |
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apply (erule (1) real_root_ge_zero) |
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apply (erule zero_le_power) |
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done |
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|
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lemma real_root_pos_unique: |
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"\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y" |
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by (erule subst, rule real_root_power_cancel) |
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|
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lemma real_root_one [simp]: "0 < n \<Longrightarrow> root n 1 = 1" |
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by (simp add: real_root_pos_unique) |
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|
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text {* Root function is strictly monotonic, hence injective *} |
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|
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lemma real_root_less_mono_lemma: |
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"\<lbrakk>0 < n; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" |
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apply (subgoal_tac "0 \<le> y") |
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apply (subgoal_tac "root n x ^ n < root n y ^ n") |
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apply (erule power_less_imp_less_base) |
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apply (erule (1) real_root_ge_zero) |
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apply simp |
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apply simp |
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done |
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|
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lemma real_root_less_mono: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y" |
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apply (cases "0 \<le> x") |
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apply (erule (2) real_root_less_mono_lemma) |
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apply (cases "0 \<le> y") |
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apply (rule_tac y=0 in order_less_le_trans) |
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apply (subgoal_tac "0 < root n (- x)") |
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apply (simp add: real_root_minus) |
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apply (simp add: real_root_gt_zero) |
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apply (simp add: real_root_ge_zero) |
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apply (subgoal_tac "root n (- y) < root n (- x)") |
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apply (simp add: real_root_minus) |
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apply (simp add: real_root_less_mono_lemma) |
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done |
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|
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lemma real_root_le_mono: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y" |
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by (auto simp add: order_le_less real_root_less_mono) |
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|
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lemma real_root_less_iff [simp]: |
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"0 < n \<Longrightarrow> (root n x < root n y) = (x < y)" |
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apply (cases "x < y") |
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apply (simp add: real_root_less_mono) |
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apply (simp add: linorder_not_less real_root_le_mono) |
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done |
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|
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lemma real_root_le_iff [simp]: |
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"0 < n \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)" |
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apply (cases "x \<le> y") |
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apply (simp add: real_root_le_mono) |
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apply (simp add: linorder_not_le real_root_less_mono) |
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done |
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|
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lemma real_root_eq_iff [simp]: |
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"0 < n \<Longrightarrow> (root n x = root n y) = (x = y)" |
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by (simp add: order_eq_iff) |
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|
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lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified] |
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lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified] |
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lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified] |
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lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified] |
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lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified] |
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|
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text {* Roots of multiplication and division *} |
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|
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lemma real_root_mult_lemma: |
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"\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> root n (x * y) = root n x * root n y" |
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by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib) |
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|
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lemma real_root_inverse_lemma: |
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"\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (inverse x) = inverse (root n x)" |
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by (simp add: real_root_pos_unique power_inverse [symmetric]) |
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295 |
|
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lemma real_root_mult: |
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assumes n: "0 < n" |
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298 |
shows "root n (x * y) = root n x * root n y" |
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proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases) |
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300 |
assume "0 \<le> x" and "0 \<le> y" |
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thus ?thesis by (rule real_root_mult_lemma [OF n]) |
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302 |
next |
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303 |
assume "0 \<le> x" and "y \<le> 0" |
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hence "0 \<le> x" and "0 \<le> - y" by simp_all |
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305 |
hence "root n (x * - y) = root n x * root n (- y)" |
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by (rule real_root_mult_lemma [OF n]) |
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thus ?thesis by (simp add: real_root_minus [OF n]) |
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308 |
next |
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309 |
assume "x \<le> 0" and "0 \<le> y" |
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hence "0 \<le> - x" and "0 \<le> y" by simp_all |
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hence "root n (- x * y) = root n (- x) * root n y" |
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by (rule real_root_mult_lemma [OF n]) |
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thus ?thesis by (simp add: real_root_minus [OF n]) |
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|
314 |
next |
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|
315 |
assume "x \<le> 0" and "y \<le> 0" |
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|
316 |
hence "0 \<le> - x" and "0 \<le> - y" by simp_all |
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|
317 |
hence "root n (- x * - y) = root n (- x) * root n (- y)" |
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|
318 |
by (rule real_root_mult_lemma [OF n]) |
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|
319 |
thus ?thesis by (simp add: real_root_minus [OF n]) |
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|
320 |
qed |
22721
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changeset
|
321 |
|
d9be18bd7a28
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|
322 |
lemma real_root_inverse: |
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|
323 |
assumes n: "0 < n" |
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changeset
|
324 |
shows "root n (inverse x) = inverse (root n x)" |
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|
325 |
proof (rule linorder_le_cases) |
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|
326 |
assume "0 \<le> x" |
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|
327 |
thus ?thesis by (rule real_root_inverse_lemma [OF n]) |
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|
328 |
next |
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|
329 |
assume "x \<le> 0" |
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|
330 |
hence "0 \<le> - x" by simp |
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|
331 |
hence "root n (inverse (- x)) = inverse (root n (- x))" |
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|
332 |
by (rule real_root_inverse_lemma [OF n]) |
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|
333 |
thus ?thesis by (simp add: real_root_minus [OF n]) |
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changeset
|
334 |
qed |
22721
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changeset
|
335 |
|
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|
336 |
lemma real_root_divide: |
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|
337 |
"0 < n \<Longrightarrow> root n (x / y) = root n x / root n y" |
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|
338 |
by (simp add: divide_inverse real_root_mult real_root_inverse) |
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changeset
|
339 |
|
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|
340 |
lemma real_root_power: |
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|
341 |
"0 < n \<Longrightarrow> root n (x ^ k) = root n x ^ k" |
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changeset
|
342 |
by (induct k, simp_all add: real_root_mult) |
22721
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moved root and sqrt lemmas from Transcendental.thy to NthRoot.thy
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diff
changeset
|
343 |
|
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changeset
|
344 |
|
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|
345 |
subsection {* Square Root *} |
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|
346 |
|
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|
347 |
definition |
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|
348 |
sqrt :: "real \<Rightarrow> real" where |
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|
349 |
"sqrt = root 2" |
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changeset
|
350 |
|
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|
351 |
lemma pos2: "0 < (2::nat)" by simp |
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changeset
|
352 |
|
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|
353 |
lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y" |
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changeset
|
354 |
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2]) |
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changeset
|
355 |
|
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|
356 |
lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>" |
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changeset
|
357 |
apply (rule real_sqrt_unique) |
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|
358 |
apply (rule power2_abs) |
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|
359 |
apply (rule abs_ge_zero) |
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|
360 |
done |
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changeset
|
361 |
|
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|
362 |
lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<twosuperior> = x" |
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changeset
|
363 |
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2]) |
22856 | 364 |
|
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changeset
|
365 |
lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)" |
22856 | 366 |
apply (rule iffI) |
367 |
apply (erule subst) |
|
368 |
apply (rule zero_le_power2) |
|
369 |
apply (erule real_sqrt_pow2) |
|
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changeset
|
370 |
done |
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changeset
|
371 |
|
22956
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|
372 |
lemma real_sqrt_zero [simp]: "sqrt 0 = 0" |
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changeset
|
373 |
unfolding sqrt_def by (rule real_root_zero) |
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changeset
|
374 |
|
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changeset
|
375 |
lemma real_sqrt_one [simp]: "sqrt 1 = 1" |
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changeset
|
376 |
unfolding sqrt_def by (rule real_root_one [OF pos2]) |
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parents:
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changeset
|
377 |
|
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changeset
|
378 |
lemma real_sqrt_minus: "sqrt (- x) = - sqrt x" |
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changeset
|
379 |
unfolding sqrt_def by (rule real_root_minus [OF pos2]) |
617140080e6a
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parents:
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changeset
|
380 |
|
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changeset
|
381 |
lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y" |
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changeset
|
382 |
unfolding sqrt_def by (rule real_root_mult [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
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diff
changeset
|
383 |
|
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parents:
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changeset
|
384 |
lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)" |
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parents:
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changeset
|
385 |
unfolding sqrt_def by (rule real_root_inverse [OF pos2]) |
617140080e6a
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huffman
parents:
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diff
changeset
|
386 |
|
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parents:
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diff
changeset
|
387 |
lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y" |
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huffman
parents:
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diff
changeset
|
388 |
unfolding sqrt_def by (rule real_root_divide [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
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diff
changeset
|
389 |
|
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huffman
parents:
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diff
changeset
|
390 |
lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k" |
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parents:
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diff
changeset
|
391 |
unfolding sqrt_def by (rule real_root_power [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
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diff
changeset
|
392 |
|
617140080e6a
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huffman
parents:
22943
diff
changeset
|
393 |
lemma real_sqrt_gt_zero: "0 < x \<Longrightarrow> 0 < sqrt x" |
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huffman
parents:
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diff
changeset
|
394 |
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2]) |
617140080e6a
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huffman
parents:
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diff
changeset
|
395 |
|
617140080e6a
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parents:
22943
diff
changeset
|
396 |
lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x" |
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huffman
parents:
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diff
changeset
|
397 |
unfolding sqrt_def by (rule real_root_ge_zero [OF pos2]) |
20687
fedb901be392
move root and sqrt stuff from Transcendental to NthRoot
huffman
parents:
20515
diff
changeset
|
398 |
|
22956
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huffman
parents:
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diff
changeset
|
399 |
lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y" |
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huffman
parents:
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diff
changeset
|
400 |
unfolding sqrt_def by (rule real_root_less_mono [OF pos2]) |
617140080e6a
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huffman
parents:
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diff
changeset
|
401 |
|
617140080e6a
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huffman
parents:
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diff
changeset
|
402 |
lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y" |
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huffman
parents:
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diff
changeset
|
403 |
unfolding sqrt_def by (rule real_root_le_mono [OF pos2]) |
617140080e6a
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huffman
parents:
22943
diff
changeset
|
404 |
|
617140080e6a
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huffman
parents:
22943
diff
changeset
|
405 |
lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)" |
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huffman
parents:
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diff
changeset
|
406 |
unfolding sqrt_def by (rule real_root_less_iff [OF pos2]) |
617140080e6a
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huffman
parents:
22943
diff
changeset
|
407 |
|
617140080e6a
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huffman
parents:
22943
diff
changeset
|
408 |
lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)" |
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huffman
parents:
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diff
changeset
|
409 |
unfolding sqrt_def by (rule real_root_le_iff [OF pos2]) |
617140080e6a
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huffman
parents:
22943
diff
changeset
|
410 |
|
617140080e6a
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huffman
parents:
22943
diff
changeset
|
411 |
lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)" |
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huffman
parents:
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diff
changeset
|
412 |
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2]) |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
413 |
|
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
414 |
lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified] |
617140080e6a
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huffman
parents:
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diff
changeset
|
415 |
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
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diff
changeset
|
416 |
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
417 |
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
418 |
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
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diff
changeset
|
419 |
|
617140080e6a
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huffman
parents:
22943
diff
changeset
|
420 |
lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
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diff
changeset
|
421 |
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
422 |
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified] |
617140080e6a
define roots of negative reals so that many lemmas no longer require side conditions; simplification solves more goals than previously
huffman
parents:
22943
diff
changeset
|
423 |
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified] |
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|
424 |
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified] |
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|
425 |
|
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|
426 |
lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" |
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|
427 |
apply auto |
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|
428 |
apply (cut_tac x = x and y = 0 in linorder_less_linear) |
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|
429 |
apply (simp add: zero_less_mult_iff) |
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|
430 |
done |
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|
431 |
|
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|
432 |
lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>" |
22856 | 433 |
apply (subst power2_eq_square [symmetric]) |
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|
434 |
apply (rule real_sqrt_abs) |
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|
435 |
done |
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|
436 |
|
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|
437 |
lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>" |
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|
438 |
by simp (* TODO: delete *) |
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|
439 |
|
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|
440 |
lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0" |
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|
441 |
by simp (* TODO: delete *) |
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|
442 |
|
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|
443 |
lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" |
22856 | 444 |
by (simp add: power_inverse [symmetric]) |
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|
445 |
|
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|
446 |
lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0" |
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|
447 |
by simp |
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|
448 |
|
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|
449 |
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x" |
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|
450 |
by simp |
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|
451 |
|
22443 | 452 |
lemma sqrt_divide_self_eq: |
453 |
assumes nneg: "0 \<le> x" |
|
454 |
shows "sqrt x / x = inverse (sqrt x)" |
|
455 |
proof cases |
|
456 |
assume "x=0" thus ?thesis by simp |
|
457 |
next |
|
458 |
assume nz: "x\<noteq>0" |
|
459 |
hence pos: "0<x" using nneg by arith |
|
460 |
show ?thesis |
|
461 |
proof (rule right_inverse_eq [THEN iffD1, THEN sym]) |
|
462 |
show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) |
|
463 |
show "inverse (sqrt x) / (sqrt x / x) = 1" |
|
464 |
by (simp add: divide_inverse mult_assoc [symmetric] |
|
465 |
power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) |
|
466 |
qed |
|
467 |
qed |
|
468 |
||
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|
469 |
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r" |
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|
470 |
apply (simp add: divide_inverse) |
d9be18bd7a28
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|
471 |
apply (case_tac "r=0") |
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changeset
|
472 |
apply (auto simp add: mult_ac) |
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|
473 |
done |
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|
474 |
|
22856 | 475 |
subsection {* Square Root of Sum of Squares *} |
476 |
||
477 |
lemma "(sqrt (x\<twosuperior> + y\<twosuperior>))\<twosuperior> = x\<twosuperior> + y\<twosuperior>" |
|
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|
478 |
by simp |
22856 | 479 |
|
480 |
lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)" |
|
481 |
by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) |
|
482 |
||
483 |
lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)" |
|
484 |
by (auto intro!: real_sqrt_ge_zero) |
|
485 |
||
486 |
lemma real_sqrt_sum_squares_mult_ge_zero [simp]: |
|
487 |
"0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))" |
|
488 |
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) |
|
489 |
||
490 |
lemma real_sqrt_sum_squares_mult_squared_eq [simp]: |
|
491 |
"sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)" |
|
22956
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|
492 |
by (auto simp add: zero_le_mult_iff) |
22856 | 493 |
|
494 |
lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)" |
|
495 |
by (rule power2_le_imp_le, simp_all) |
|
496 |
||
497 |
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(x\<twosuperior> + y\<twosuperior>)" |
|
498 |
by (rule power2_le_imp_le, simp_all) |
|
499 |
||
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|
500 |
lemma real_sqrt_sos_less_one_iff: "(sqrt (x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)" |
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|
501 |
by (rule real_sqrt_lt_1_iff) |
22856 | 502 |
|
22956
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|
503 |
lemma real_sqrt_sos_eq_one_iff: "(sqrt (x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)" |
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|
504 |
by (rule real_sqrt_eq_1_iff) |
22721
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|
505 |
|
22858 | 506 |
lemma power2_sum: |
507 |
fixes x y :: "'a::{number_ring,recpower}" |
|
508 |
shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y" |
|
509 |
by (simp add: left_distrib right_distrib power2_eq_square) |
|
510 |
||
511 |
lemma power2_diff: |
|
512 |
fixes x y :: "'a::{number_ring,recpower}" |
|
513 |
shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y" |
|
514 |
by (simp add: left_diff_distrib right_diff_distrib power2_eq_square) |
|
515 |
||
516 |
lemma real_sqrt_sum_squares_triangle_ineq: |
|
517 |
"sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)" |
|
518 |
apply (rule power2_le_imp_le, simp) |
|
519 |
apply (simp add: power2_sum) |
|
520 |
apply (simp only: mult_assoc right_distrib [symmetric]) |
|
521 |
apply (rule mult_left_mono) |
|
522 |
apply (rule power2_le_imp_le) |
|
523 |
apply (simp add: power2_sum power_mult_distrib) |
|
524 |
apply (simp add: ring_distrib) |
|
525 |
apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior> - 2 * (a * c) * (b * d)", simp) |
|
526 |
apply (rule_tac b="(a * d - b * c)\<twosuperior>" in ord_le_eq_trans) |
|
527 |
apply (rule zero_le_power2) |
|
528 |
apply (simp add: power2_diff power_mult_distrib) |
|
529 |
apply (simp add: mult_nonneg_nonneg) |
|
530 |
apply simp |
|
531 |
apply (simp add: add_increasing) |
|
532 |
done |
|
533 |
||
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changeset
|
534 |
text "Legacy theorem names:" |
617140080e6a
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huffman
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changeset
|
535 |
lemmas real_root_pos2 = real_root_power_cancel |
617140080e6a
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huffman
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changeset
|
536 |
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le] |
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huffman
parents:
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changeset
|
537 |
lemmas real_root_pos_pos_le = real_root_ge_zero |
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huffman
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changeset
|
538 |
lemmas real_sqrt_mult_distrib = real_sqrt_mult |
617140080e6a
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huffman
parents:
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changeset
|
539 |
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult |
617140080e6a
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huffman
parents:
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changeset
|
540 |
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff |
617140080e6a
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huffman
parents:
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changeset
|
541 |
|
617140080e6a
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huffman
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changeset
|
542 |
(* needed for CauchysMeanTheorem.het_base from AFP *) |
617140080e6a
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huffman
parents:
22943
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changeset
|
543 |
lemma real_root_pos: "0 < x \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x" |
617140080e6a
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huffman
parents:
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changeset
|
544 |
by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le]) |
617140080e6a
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huffman
parents:
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diff
changeset
|
545 |
|
617140080e6a
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huffman
parents:
22943
diff
changeset
|
546 |
(* FIXME: the stronger version of real_root_less_iff |
617140080e6a
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huffman
parents:
22943
diff
changeset
|
547 |
breaks CauchysMeanTheorem.list_gmean_gt_iff from AFP. *) |
617140080e6a
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huffman
parents:
22943
diff
changeset
|
548 |
|
617140080e6a
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huffman
parents:
22943
diff
changeset
|
549 |
declare real_root_less_iff [simp del] |
617140080e6a
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huffman
parents:
22943
diff
changeset
|
550 |
lemma real_root_less_iff_nonneg [simp]: |
617140080e6a
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huffman
parents:
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changeset
|
551 |
"\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> (root n x < root n y) = (x < y)" |
617140080e6a
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huffman
parents:
22943
diff
changeset
|
552 |
by (rule real_root_less_iff) |
617140080e6a
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huffman
parents:
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changeset
|
553 |
|
14324 | 554 |
end |