paulson@12196: (* Title : NthRoot.thy paulson@12196: Author : Jacques D. Fleuriot paulson@12196: Copyright : 1998 University of Cambridge paulson@14477: Conversion to Isar and new proofs by Lawrence C Paulson, 2004 paulson@12196: *) paulson@12196: paulson@14324: header{*Existence of Nth Root*} paulson@14324: nipkow@15131: theory NthRoot huffman@21865: imports SEQ Parity nipkow@15131: begin paulson@14324: huffman@20687: definition wenzelm@21404: root :: "[nat, real] \ real" where huffman@20687: "root n x = (THE u. (0 < x \ 0 < u) \ (u ^ n = x))" huffman@20687: wenzelm@21404: definition wenzelm@21404: sqrt :: "real \ real" where huffman@20687: "sqrt x = root 2 x" huffman@20687: huffman@20687: wenzelm@14767: text {* wenzelm@14767: Various lemmas needed for this result. We follow the proof given by wenzelm@14767: John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis wenzelm@14767: Webnotes available at \url{http://www.math.unl.edu/~webnotes}. wenzelm@14767: wenzelm@14767: Lemmas about sequences of reals are used to reach the result. wenzelm@14767: *} paulson@14324: paulson@14324: lemma lemma_nth_realpow_non_empty: paulson@14324: "[| (0::real) < a; 0 < n |] ==> \s. s : {x. x ^ n <= a & 0 < x}" paulson@14324: apply (case_tac "1 <= a") paulson@14477: apply (rule_tac x = 1 in exI) paulson@14334: apply (drule_tac [2] linorder_not_le [THEN iffD1]) paulson@14477: apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp) paulson@14348: apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc) paulson@14324: done paulson@14324: paulson@14348: text{*Used only just below*} paulson@14348: lemma realpow_ge_self2: "[| (1::real) \ r; 0 < n |] ==> r \ r ^ n" paulson@14348: by (insert power_increasing [of 1 n r], simp) paulson@14348: paulson@14324: lemma lemma_nth_realpow_isUb_ex: paulson@14324: "[| (0::real) < a; 0 < n |] paulson@14324: ==> \u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u" paulson@14324: apply (case_tac "1 <= a") paulson@14477: apply (rule_tac x = a in exI) paulson@14334: apply (drule_tac [2] linorder_not_le [THEN iffD1]) paulson@14477: apply (rule_tac [2] x = 1 in exI) paulson@14477: apply (rule_tac [!] setleI [THEN isUbI], safe) paulson@14324: apply (simp_all (no_asm)) paulson@14324: apply (rule_tac [!] ccontr) paulson@14334: apply (drule_tac [!] linorder_not_le [THEN iffD1]) paulson@14477: apply (drule realpow_ge_self2, assumption) paulson@14477: apply (drule_tac n = n in realpow_less) paulson@14324: apply (assumption+) paulson@14477: apply (drule real_le_trans, assumption) paulson@14477: apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp) paulson@14477: apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto) paulson@14324: done paulson@14324: paulson@14324: lemma nth_realpow_isLub_ex: paulson@14324: "[| (0::real) < a; 0 < n |] paulson@14324: ==> \u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u" paulson@14365: by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete) paulson@14365: paulson@14324: paulson@14324: subsection{*First Half -- Lemmas First*} paulson@14324: paulson@14324: lemma lemma_nth_realpow_seq: paulson@14324: "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u paulson@14324: ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}" paulson@14477: apply (safe, drule isLubD2, blast) paulson@14365: apply (simp add: linorder_not_less [symmetric]) paulson@14324: done paulson@14324: paulson@14324: lemma lemma_nth_realpow_isLub_gt_zero: paulson@14324: "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; paulson@14324: 0 < a; 0 < n |] ==> 0 < u" paulson@14477: apply (drule lemma_nth_realpow_non_empty, auto) paulson@14477: apply (drule_tac y = s in isLub_isUb [THEN isUbD]) paulson@14324: apply (auto intro: order_less_le_trans) paulson@14324: done paulson@14324: paulson@14324: lemma lemma_nth_realpow_isLub_ge: paulson@14324: "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; paulson@14324: 0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n" paulson@14477: apply safe paulson@14477: apply (frule lemma_nth_realpow_seq, safe) paulson@15085: apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric] paulson@15085: iff: real_0_less_add_iff) --{*legacy iff rule!*} paulson@14365: apply (simp add: linorder_not_less) paulson@14324: apply (rule order_less_trans [of _ 0]) paulson@14325: apply (auto intro: lemma_nth_realpow_isLub_gt_zero) paulson@14324: done paulson@14324: paulson@14324: text{*First result we want*} paulson@14324: lemma realpow_nth_ge: paulson@14324: "[| (0::real) < a; 0 < n; paulson@14324: isLub (UNIV::real set) paulson@14324: {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n" paulson@14477: apply (frule lemma_nth_realpow_isLub_ge, safe) paulson@14324: apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const]) paulson@14334: apply (auto simp add: real_of_nat_def) paulson@14324: done paulson@14324: paulson@14324: subsection{*Second Half*} paulson@14324: paulson@14324: lemma less_isLub_not_isUb: paulson@14324: "[| isLub (UNIV::real set) S u; x < u |] paulson@14324: ==> ~ isUb (UNIV::real set) S x" paulson@14477: apply safe paulson@14477: apply (drule isLub_le_isUb, assumption) paulson@14477: apply (drule order_less_le_trans, auto) paulson@14324: done paulson@14324: paulson@14324: lemma not_isUb_less_ex: paulson@14324: "~ isUb (UNIV::real set) S u ==> \x \ S. u < x" wenzelm@18585: apply (rule ccontr, erule contrapos_np) paulson@14324: apply (rule setleI [THEN isUbI]) paulson@14365: apply (auto simp add: linorder_not_less [symmetric]) paulson@14324: done paulson@14324: paulson@14325: lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r" paulson@14334: apply (simp (no_asm) add: right_distrib) paulson@14334: apply (rule add_less_cancel_left [of "-r", THEN iffD1]) avigad@16775: apply (auto intro: mult_pos_pos paulson@14334: simp add: add_assoc [symmetric] neg_less_0_iff_less) paulson@14325: done paulson@14325: paulson@14325: lemma real_mult_add_one_minus_ge_zero: paulson@14325: "0 < r ==> 0 <= r*(1 + -inverse(real (Suc n)))" paulson@15085: by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff) paulson@14325: paulson@14324: lemma lemma_nth_realpow_isLub_le: paulson@14324: "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u; paulson@14325: 0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a" paulson@14477: apply safe paulson@14324: apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex]) paulson@14477: apply (rule_tac n = k in real_mult_less_self) paulson@14477: apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe) paulson@14477: apply (drule_tac n = k in paulson@14477: lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+) paulson@14348: apply (blast intro: order_trans order_less_imp_le power_mono) paulson@14324: done paulson@14324: paulson@14324: text{*Second result we want*} paulson@14324: lemma realpow_nth_le: paulson@14324: "[| (0::real) < a; 0 < n; paulson@14324: isLub (UNIV::real set) paulson@14324: {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a" paulson@14477: apply (frule lemma_nth_realpow_isLub_le, safe) paulson@14348: apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult paulson@14348: [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2]) paulson@14334: apply (auto simp add: real_of_nat_def) paulson@14324: done paulson@14324: paulson@14348: text{*The theorem at last!*} paulson@14324: lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \r. r ^ n = a" paulson@14477: apply (frule nth_realpow_isLub_ex, auto) paulson@14477: apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym) paulson@14324: done paulson@14324: paulson@14324: (* positive only *) paulson@14324: lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \r. 0 < r & r ^ n = a" paulson@14477: apply (frule nth_realpow_isLub_ex, auto) paulson@14477: apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero) paulson@14324: done paulson@14324: paulson@14324: lemma realpow_pos_nth2: "(0::real) < a ==> \r. 0 < r & r ^ Suc n = a" paulson@14477: by (blast intro: realpow_pos_nth) paulson@14324: paulson@14324: (* uniqueness of nth positive root *) paulson@14324: lemma realpow_pos_nth_unique: paulson@14324: "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a" paulson@14324: apply (auto intro!: realpow_pos_nth) paulson@14477: apply (cut_tac x = r and y = y in linorder_less_linear, auto) paulson@14477: apply (drule_tac x = r in realpow_less) paulson@14477: apply (drule_tac [4] x = y in realpow_less, auto) paulson@14324: done paulson@14324: huffman@20687: subsection {* Nth Root *} huffman@20687: huffman@20687: lemma real_root_zero [simp]: "root (Suc n) 0 = 0" huffman@20687: apply (simp add: root_def) huffman@20687: apply (safe intro!: the_equality power_0_Suc elim!: realpow_zero_zero) huffman@20687: done huffman@20687: huffman@20687: lemma real_root_pow_pos: huffman@20687: "0 < x ==> (root (Suc n) x) ^ (Suc n) = x" huffman@20687: apply (simp add: root_def del: realpow_Suc) huffman@20687: apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp) huffman@20687: apply (erule theI' [THEN conjunct2]) huffman@20687: done huffman@20687: huffman@20687: lemma real_root_pow_pos2: "0 \ x ==> (root (Suc n) x) ^ (Suc n) = x" huffman@20687: by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos) huffman@20687: huffman@20687: lemma real_root_pos: huffman@20687: "0 < x ==> root(Suc n) (x ^ (Suc n)) = x" huffman@20687: apply (simp add: root_def) huffman@20687: apply (rule the_equality) huffman@20687: apply (frule_tac [2] n = n in zero_less_power) huffman@20687: apply (auto simp add: zero_less_mult_iff) huffman@20687: apply (rule_tac x = u and y = x in linorder_cases) huffman@20687: apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less]) huffman@20687: apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less]) huffman@20687: apply (auto) huffman@20687: done huffman@20687: huffman@20687: lemma real_root_pos2: "0 \ x ==> root(Suc n) (x ^ (Suc n)) = x" huffman@20687: by (auto dest!: real_le_imp_less_or_eq real_root_pos) huffman@20687: huffman@20687: lemma real_root_gt_zero: huffman@20687: "0 < x ==> 0 < root (Suc n) x" huffman@20687: apply (simp add: root_def del: realpow_Suc) huffman@20687: apply (drule_tac n="Suc n" in realpow_pos_nth_unique, simp) huffman@20687: apply (erule theI' [THEN conjunct1]) huffman@20687: done huffman@20687: huffman@20687: lemma real_root_pos_pos: huffman@20687: "0 < x ==> 0 \ root(Suc n) x" huffman@20687: by (rule real_root_gt_zero [THEN order_less_imp_le]) huffman@20687: huffman@20687: lemma real_root_pos_pos_le: "0 \ x ==> 0 \ root(Suc n) x" huffman@20687: by (auto simp add: order_le_less real_root_gt_zero) huffman@20687: huffman@20687: lemma real_root_one [simp]: "root (Suc n) 1 = 1" huffman@20687: apply (simp add: root_def) huffman@20687: apply (rule the_equality, auto) huffman@20687: apply (rule ccontr) huffman@20687: apply (rule_tac x = u and y = 1 in linorder_cases) huffman@20687: apply (drule_tac n = n in realpow_Suc_less_one) huffman@20687: apply (drule_tac [4] n = n in power_gt1_lemma) huffman@20687: apply (auto) huffman@20687: done huffman@20687: huffman@20687: huffman@20687: subsection{*Square Root*} huffman@20687: huffman@20687: text{*needed because 2 is a binary numeral!*} huffman@20687: lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))" huffman@20687: by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 huffman@20687: add: nat_numeral_0_eq_0 [symmetric]) huffman@20687: huffman@20687: lemma real_sqrt_zero [simp]: "sqrt 0 = 0" huffman@20687: by (simp add: sqrt_def) huffman@20687: huffman@20687: lemma real_sqrt_one [simp]: "sqrt 1 = 1" huffman@20687: by (simp add: sqrt_def) huffman@20687: huffman@20687: lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\ = x) = (0 \ x)" huffman@20687: apply (simp add: sqrt_def) huffman@20687: apply (rule iffI) huffman@20687: apply (cut_tac r = "root 2 x" in realpow_two_le) huffman@20687: apply (simp add: numeral_2_eq_2) huffman@20687: apply (subst numeral_2_eq_2) huffman@20687: apply (erule real_root_pow_pos2) huffman@20687: done huffman@20687: huffman@20687: lemma [simp]: "(sqrt(u2\ + v2\))\ = u2\ + v2\" huffman@20687: by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]]) huffman@20687: huffman@20687: lemma real_sqrt_pow2 [simp]: "0 \ x ==> (sqrt x)\ = x" huffman@20687: by (simp) huffman@20687: huffman@20687: lemma real_sqrt_abs_abs [simp]: "sqrt\x\ ^ 2 = \x\" huffman@20687: by (rule real_sqrt_pow2_iff [THEN iffD2], arith) huffman@20687: huffman@20687: lemma real_pow_sqrt_eq_sqrt_pow: huffman@20687: "0 \ x ==> (sqrt x)\ = sqrt(x\)" huffman@20687: apply (simp add: sqrt_def) huffman@20687: apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2) huffman@20687: done huffman@20687: huffman@20687: lemma real_pow_sqrt_eq_sqrt_abs_pow2: huffman@20687: "0 \ x ==> (sqrt x)\ = sqrt(\x\ ^ 2)" huffman@20687: by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric]) huffman@20687: huffman@20687: lemma real_sqrt_pow_abs: "0 \ x ==> (sqrt x)\ = \x\" huffman@20687: apply (rule real_sqrt_abs_abs [THEN subst]) huffman@20687: apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst]) huffman@20687: apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric]) huffman@20687: apply (assumption, arith) huffman@20687: done huffman@20687: huffman@20687: lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)" huffman@20687: apply auto huffman@20687: apply (cut_tac x = x and y = 0 in linorder_less_linear) huffman@20687: apply (simp add: zero_less_mult_iff) huffman@20687: done huffman@20687: huffman@20687: lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)" huffman@20687: by (simp add: sqrt_def real_root_gt_zero) huffman@20687: huffman@20687: lemma real_sqrt_ge_zero: "0 \ x ==> 0 \ sqrt(x)" huffman@20687: by (auto intro: real_sqrt_gt_zero simp add: order_le_less) huffman@20687: huffman@20687: lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \ sqrt(x*x + y*y)" huffman@20687: by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) huffman@20687: huffman@20687: huffman@20687: (*we need to prove something like this: huffman@20687: lemma "[|r ^ n = a; 0 0 < r|] ==> root n a = r" huffman@20687: apply (case_tac n, simp) huffman@20687: apply (simp add: root_def) huffman@20687: apply (rule someI2 [of _ r], safe) huffman@20687: apply (auto simp del: realpow_Suc dest: power_inject_base) huffman@20687: *) huffman@20687: huffman@20687: lemma sqrt_eqI: "[|r\ = a; 0 \ r|] ==> sqrt a = r" huffman@20687: apply (erule subst) huffman@20687: apply (simp add: sqrt_def numeral_2_eq_2 del: realpow_Suc) huffman@20687: apply (erule real_root_pos2) huffman@20687: done huffman@20687: huffman@20687: lemma real_sqrt_mult_distrib: huffman@20687: "[| 0 \ x; 0 \ y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" huffman@20687: apply (rule sqrt_eqI) huffman@20687: apply (simp add: power_mult_distrib) huffman@20687: apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) huffman@20687: done huffman@20687: huffman@20687: lemma real_sqrt_mult_distrib2: huffman@20687: "[|0\x; 0\y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)" huffman@20687: by (auto intro: real_sqrt_mult_distrib simp add: order_le_less) huffman@20687: huffman@20687: lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \ sqrt (x\ + y\)" huffman@20687: by (auto intro!: real_sqrt_ge_zero) huffman@20687: huffman@20687: lemma real_sqrt_sum_squares_mult_ge_zero [simp]: huffman@20687: "0 \ sqrt ((x\ + y\)*(xa\ + ya\))" huffman@20687: by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff) huffman@20687: huffman@20687: lemma real_sqrt_sum_squares_mult_squared_eq [simp]: huffman@20687: "sqrt ((x\ + y\) * (xa\ + ya\)) ^ 2 = (x\ + y\) * (xa\ + ya\)" huffman@20687: by (auto simp add: zero_le_mult_iff simp del: realpow_Suc) huffman@20687: huffman@20687: lemma real_sqrt_abs [simp]: "sqrt(x\) = \x\" huffman@20687: apply (rule abs_realpow_two [THEN subst]) huffman@20687: apply (rule real_sqrt_abs_abs [THEN subst]) huffman@20687: apply (subst real_pow_sqrt_eq_sqrt_pow) huffman@20687: apply (auto simp add: numeral_2_eq_2) huffman@20687: done huffman@20687: huffman@20687: lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \x\" huffman@20687: apply (rule realpow_two [THEN subst]) huffman@20687: apply (subst numeral_2_eq_2 [symmetric]) huffman@20687: apply (rule real_sqrt_abs) huffman@20687: done huffman@20687: huffman@20687: lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\" huffman@20687: by simp huffman@20687: huffman@20687: lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \ 0" huffman@20687: apply (frule real_sqrt_pow2_gt_zero) huffman@20687: apply (auto simp add: numeral_2_eq_2) huffman@20687: done huffman@20687: huffman@20687: lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x" wenzelm@20898: by (cut_tac n = 2 and a = "sqrt x" in power_inverse [symmetric], auto) huffman@20687: huffman@20687: lemma real_sqrt_eq_zero_cancel: "[| 0 \ x; sqrt(x) = 0|] ==> x = 0" huffman@20687: apply (drule real_le_imp_less_or_eq) huffman@20687: apply (auto dest: real_sqrt_not_eq_zero) huffman@20687: done huffman@20687: huffman@20687: lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \ x ==> ((sqrt x = 0) = (x=0))" huffman@20687: by (auto simp add: real_sqrt_eq_zero_cancel) huffman@20687: huffman@20687: lemma real_sqrt_sum_squares_ge1 [simp]: "x \ sqrt(x\ + y\)" huffman@20687: apply (subgoal_tac "x \ 0 | 0 \ x", safe) huffman@20687: apply (rule real_le_trans) huffman@20687: apply (auto simp del: realpow_Suc) huffman@20687: apply (rule_tac n = 1 in realpow_increasing) huffman@20687: apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc) huffman@20687: done huffman@20687: huffman@20687: lemma real_sqrt_sum_squares_ge2 [simp]: "y \ sqrt(z\ + y\)" huffman@20687: apply (simp (no_asm) add: real_add_commute del: realpow_Suc) huffman@20687: done huffman@20687: huffman@20687: lemma real_sqrt_ge_one: "1 \ x ==> 1 \ sqrt x" huffman@20687: apply (rule_tac n = 1 in realpow_increasing) huffman@20687: apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp huffman@20687: del: realpow_Suc) huffman@20687: done huffman@20687: paulson@14324: end