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theory Transcendental(* Title : Transcendental.thy Author : Jacques D. Fleuriot Copyright : 1998,1999 University of Cambridge 1999,2001 University of Edinburgh Conversion to Isar and new proofs by Lawrence C Paulson, 2004 *) header{*Power Series, Transcendental Functions etc.*} theory Transcendental imports NthRoot Fact Series EvenOdd Deriv begin subsection{*Properties of Power Series*} lemma lemma_realpow_diff: fixes y :: "'a::recpower" shows "p ≤ n ==> y ^ (Suc n - p) = (y ^ (n - p)) * y" proof - assume "p ≤ n" hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le) thus ?thesis by (simp add: power_Suc power_commutes) qed lemma lemma_realpow_diff_sumr: fixes y :: "'a::{recpower,comm_semiring_0}" shows "(∑p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) = y * (∑p=0..<Suc n. (x ^ p) * y ^ (n - p))" by (auto simp add: setsum_right_distrib lemma_realpow_diff mult_ac simp del: setsum_op_ivl_Suc cong: strong_setsum_cong) lemma lemma_realpow_diff_sumr2: fixes y :: "'a::{recpower,comm_ring}" shows "x ^ (Suc n) - y ^ (Suc n) = (x - y) * (∑p=0..<Suc n. (x ^ p) * y ^ (n - p))" apply (induct n, simp add: power_Suc) apply (simp add: power_Suc del: setsum_op_ivl_Suc) apply (subst setsum_op_ivl_Suc) apply (subst lemma_realpow_diff_sumr) apply (simp add: right_distrib del: setsum_op_ivl_Suc) apply (subst mult_left_commute [where a="x - y"]) apply (erule subst) apply (simp add: power_Suc ring_simps) done lemma lemma_realpow_rev_sumr: "(∑p=0..<Suc n. (x ^ p) * (y ^ (n - p))) = (∑p=0..<Suc n. (x ^ (n - p)) * (y ^ p))" apply (rule setsum_reindex_cong [where f="λi. n - i"]) apply (rule inj_onI, simp) apply auto apply (rule_tac x="n - x" in image_eqI, simp, simp) done text{*Power series has a `circle` of convergence, i.e. if it sums for @{term x}, then it sums absolutely for @{term z} with @{term "¦z¦ < ¦x¦"}.*} lemma powser_insidea: fixes x z :: "'a::{real_normed_field,banach,recpower}" assumes 1: "summable (λn. f n * x ^ n)" assumes 2: "norm z < norm x" shows "summable (λn. norm (f n * z ^ n))" proof - from 2 have x_neq_0: "x ≠ 0" by clarsimp from 1 have "(λn. f n * x ^ n) ----> 0" by (rule summable_LIMSEQ_zero) hence "convergent (λn. f n * x ^ n)" by (rule convergentI) hence "Cauchy (λn. f n * x ^ n)" by (simp add: Cauchy_convergent_iff) hence "Bseq (λn. f n * x ^ n)" by (rule Cauchy_Bseq) then obtain K where 3: "0 < K" and 4: "∀n. norm (f n * x ^ n) ≤ K" by (simp add: Bseq_def, safe) have "∃N. ∀n≥N. norm (norm (f n * z ^ n)) ≤ K * norm (z ^ n) * inverse (norm (x ^ n))" proof (intro exI allI impI) fix n::nat assume "0 ≤ n" have "norm (norm (f n * z ^ n)) * norm (x ^ n) = norm (f n * x ^ n) * norm (z ^ n)" by (simp add: norm_mult abs_mult) also have "… ≤ K * norm (z ^ n)" by (simp only: mult_right_mono 4 norm_ge_zero) also have "… = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))" by (simp add: x_neq_0) also have "… = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)" by (simp only: mult_assoc) finally show "norm (norm (f n * z ^ n)) ≤ K * norm (z ^ n) * inverse (norm (x ^ n))" by (simp add: mult_le_cancel_right x_neq_0) qed moreover have "summable (λn. K * norm (z ^ n) * inverse (norm (x ^ n)))" proof - from 2 have "norm (norm (z * inverse x)) < 1" using x_neq_0 by (simp add: nonzero_norm_divide divide_inverse [symmetric]) hence "summable (λn. norm (z * inverse x) ^ n)" by (rule summable_geometric) hence "summable (λn. K * norm (z * inverse x) ^ n)" by (rule summable_mult) thus "summable (λn. K * norm (z ^ n) * inverse (norm (x ^ n)))" using x_neq_0 by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib power_inverse norm_power mult_assoc) qed ultimately show "summable (λn. norm (f n * z ^ n))" by (rule summable_comparison_test) qed lemma powser_inside: fixes f :: "nat => 'a::{real_normed_field,banach,recpower}" shows "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |] ==> summable (%n. f(n) * (z ^ n))" by (rule powser_insidea [THEN summable_norm_cancel]) subsection{*Term-by-Term Differentiability of Power Series*} definition diffs :: "(nat => 'a::ring_1) => nat => 'a" where "diffs c = (%n. of_nat (Suc n) * c(Suc n))" text{*Lemma about distributing negation over it*} lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)" by (simp add: diffs_def) text{*Show that we can shift the terms down one*} lemma lemma_diffs: "(∑n=0..<n. (diffs c)(n) * (x ^ n)) = (∑n=0..<n. of_nat n * c(n) * (x ^ (n - Suc 0))) + (of_nat n * c(n) * x ^ (n - Suc 0))" apply (induct "n") apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def) done lemma lemma_diffs2: "(∑n=0..<n. of_nat n * c(n) * (x ^ (n - Suc 0))) = (∑n=0..<n. (diffs c)(n) * (x ^ n)) - (of_nat n * c(n) * x ^ (n - Suc 0))" by (auto simp add: lemma_diffs) lemma diffs_equiv: "summable (%n. (diffs c)(n) * (x ^ n)) ==> (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums (∑n. (diffs c)(n) * (x ^ n))" apply (subgoal_tac " (%n. of_nat n * c (n) * (x ^ (n - Suc 0))) ----> 0") apply (rule_tac [2] LIMSEQ_imp_Suc) apply (drule summable_sums) apply (auto simp add: sums_def) apply (drule_tac X="(λn. ∑n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff) apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric]) apply (simp add: diffs_def summable_LIMSEQ_zero) done lemma lemma_termdiff1: fixes z :: "'a :: {recpower,comm_ring}" shows "(∑p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) = (∑p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))" by (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac cong: strong_setsum_cong) lemma less_add_one: "m < n ==> (∃d. n = m + d + Suc 0)" by (simp add: less_iff_Suc_add) lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)" by arith lemma sumr_diff_mult_const2: "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (∑i = 0..<n. f i - r)" by (simp add: setsum_subtractf) lemma lemma_termdiff2: fixes h :: "'a :: {recpower,field}" assumes h: "h ≠ 0" shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) = h * (∑p=0..< n - Suc 0. ∑q=0..< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs") apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h) apply (simp add: right_diff_distrib diff_divide_distrib h) apply (simp add: mult_assoc [symmetric]) apply (cases "n", simp) apply (simp add: lemma_realpow_diff_sumr2 h right_diff_distrib [symmetric] mult_assoc del: realpow_Suc setsum_op_ivl_Suc of_nat_Suc) apply (subst lemma_realpow_rev_sumr) apply (subst sumr_diff_mult_const2) apply simp apply (simp only: lemma_termdiff1 setsum_right_distrib) apply (rule setsum_cong [OF refl]) apply (simp add: diff_minus [symmetric] less_iff_Suc_add) apply (clarify) apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac del: setsum_op_ivl_Suc realpow_Suc) apply (subst mult_assoc [symmetric], subst power_add [symmetric]) apply (simp add: mult_ac) done lemma real_setsum_nat_ivl_bounded2: fixes K :: "'a::ordered_semidom" assumes f: "!!p::nat. p < n ==> f p ≤ K" assumes K: "0 ≤ K" shows "setsum f {0..<n-k} ≤ of_nat n * K" apply (rule order_trans [OF setsum_mono]) apply (rule f, simp) apply (simp add: mult_right_mono K) done lemma lemma_termdiff3: fixes h z :: "'a::{real_normed_field,recpower}" assumes 1: "h ≠ 0" assumes 2: "norm z ≤ K" assumes 3: "norm (z + h) ≤ K" shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) ≤ of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" proof - have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) = norm (∑p = 0..<n - Suc 0. ∑q = 0..<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) * norm h" apply (subst lemma_termdiff2 [OF 1]) apply (subst norm_mult) apply (rule mult_commute) done also have "… ≤ of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h" proof (rule mult_right_mono [OF _ norm_ge_zero]) from norm_ge_zero 2 have K: "0 ≤ K" by (rule order_trans) have le_Kn: "!!i j n. i + j = n ==> norm ((z + h) ^ i * z ^ j) ≤ K ^ n" apply (erule subst) apply (simp only: norm_mult norm_power power_add) apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K) done show "norm (∑p = 0..<n - Suc 0. ∑q = 0..<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) ≤ of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))" apply (intro order_trans [OF norm_setsum] real_setsum_nat_ivl_bounded2 mult_nonneg_nonneg zero_le_imp_of_nat zero_le_power K) apply (rule le_Kn, simp) done qed also have "… = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" by (simp only: mult_assoc) finally show ?thesis . qed lemma lemma_termdiff4: fixes f :: "'a::{real_normed_field,recpower} => 'b::real_normed_vector" assumes k: "0 < (k::real)" assumes le: "!!h. [|h ≠ 0; norm h < k|] ==> norm (f h) ≤ K * norm h" shows "f -- 0 --> 0" proof (simp add: LIM_def, safe) fix r::real assume r: "0 < r" have zero_le_K: "0 ≤ K" apply (cut_tac k) apply (cut_tac h="of_real (k/2)" in le, simp) apply (simp del: of_real_divide) apply (drule order_trans [OF norm_ge_zero]) apply (simp add: zero_le_mult_iff) done show "∃s. 0 < s ∧ (∀x. x ≠ 0 ∧ norm x < s --> norm (f x) < r)" proof (cases) assume "K = 0" with k r le have "0 < k ∧ (∀x. x ≠ 0 ∧ norm x < k --> norm (f x) < r)" by simp thus "∃s. 0 < s ∧ (∀x. x ≠ 0 ∧ norm x < s --> norm (f x) < r)" .. next assume K_neq_zero: "K ≠ 0" with zero_le_K have K: "0 < K" by simp show "∃s. 0 < s ∧ (∀x. x ≠ 0 ∧ norm x < s --> norm (f x) < r)" proof (rule exI, safe) from k r K show "0 < min k (r * inverse K / 2)" by (simp add: mult_pos_pos positive_imp_inverse_positive) next fix x::'a assume x1: "x ≠ 0" and x2: "norm x < min k (r * inverse K / 2)" from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2" by simp_all from x1 x3 le have "norm (f x) ≤ K * norm x" by simp also from x4 K have "K * norm x < K * (r * inverse K / 2)" by (rule mult_strict_left_mono) also have "… = r / 2" using K_neq_zero by simp also have "r / 2 < r" using r by simp finally show "norm (f x) < r" . qed qed qed lemma lemma_termdiff5: fixes g :: "'a::{recpower,real_normed_field} => nat => 'b::banach" assumes k: "0 < (k::real)" assumes f: "summable f" assumes le: "!!h n. [|h ≠ 0; norm h < k|] ==> norm (g h n) ≤ f n * norm h" shows "(λh. suminf (g h)) -- 0 --> 0" proof (rule lemma_termdiff4 [OF k]) fix h::'a assume "h ≠ 0" and "norm h < k" hence A: "∀n. norm (g h n) ≤ f n * norm h" by (simp add: le) hence "∃N. ∀n≥N. norm (norm (g h n)) ≤ f n * norm h" by simp moreover from f have B: "summable (λn. f n * norm h)" by (rule summable_mult2) ultimately have C: "summable (λn. norm (g h n))" by (rule summable_comparison_test) hence "norm (suminf (g h)) ≤ (∑n. norm (g h n))" by (rule summable_norm) also from A C B have "(∑n. norm (g h n)) ≤ (∑n. f n * norm h)" by (rule summable_le) also from f have "(∑n. f n * norm h) = suminf f * norm h" by (rule suminf_mult2 [symmetric]) finally show "norm (suminf (g h)) ≤ suminf f * norm h" . qed text{* FIXME: Long proofs*} lemma termdiffs_aux: fixes x :: "'a::{recpower,real_normed_field,banach}" assumes 1: "summable (λn. diffs (diffs c) n * K ^ n)" assumes 2: "norm x < norm K" shows "(λh. ∑n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" proof - from dense [OF 2] obtain r where r1: "norm x < r" and r2: "r < norm K" by fast from norm_ge_zero r1 have r: "0 < r" by (rule order_le_less_trans) hence r_neq_0: "r ≠ 0" by simp show ?thesis proof (rule lemma_termdiff5) show "0 < r - norm x" using r1 by simp next from r r2 have "norm (of_real r::'a) < norm K" by simp with 1 have "summable (λn. norm (diffs (diffs c) n * (of_real r ^ n)))" by (rule powser_insidea) hence "summable (λn. diffs (diffs (λn. norm (c n))) n * r ^ n)" using r by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc) hence "summable (λn. of_nat n * diffs (λn. norm (c n)) n * r ^ (n - Suc 0))" by (rule diffs_equiv [THEN sums_summable]) also have "(λn. of_nat n * diffs (λn. norm (c n)) n * r ^ (n - Suc 0)) = (λn. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))" apply (rule ext) apply (simp add: diffs_def) apply (case_tac n, simp_all add: r_neq_0) done finally have "summable (λn. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))" by (rule diffs_equiv [THEN sums_summable]) also have "(λn. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) = (λn. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" apply (rule ext) apply (case_tac "n", simp) apply (case_tac "nat", simp) apply (simp add: r_neq_0) done finally show "summable (λn. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" . next fix h::'a and n::nat assume h: "h ≠ 0" assume "norm h < r - norm x" hence "norm x + norm h < r" by simp with norm_triangle_ineq have xh: "norm (x + h) < r" by (rule order_le_less_trans) show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) ≤ norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h" apply (simp only: norm_mult mult_assoc) apply (rule mult_left_mono [OF _ norm_ge_zero]) apply (simp (no_asm) add: mult_assoc [symmetric]) apply (rule lemma_termdiff3) apply (rule h) apply (rule r1 [THEN order_less_imp_le]) apply (rule xh [THEN order_less_imp_le]) done qed qed lemma termdiffs: fixes K x :: "'a::{recpower,real_normed_field,banach}" assumes 1: "summable (λn. c n * K ^ n)" assumes 2: "summable (λn. (diffs c) n * K ^ n)" assumes 3: "summable (λn. (diffs (diffs c)) n * K ^ n)" assumes 4: "norm x < norm K" shows "DERIV (λx. ∑n. c n * x ^ n) x :> (∑n. (diffs c) n * x ^ n)" proof (simp add: deriv_def, rule LIM_zero_cancel) show "(λh. (suminf (λn. c n * (x + h) ^ n) - suminf (λn. c n * x ^ n)) / h - suminf (λn. diffs c n * x ^ n)) -- 0 --> 0" proof (rule LIM_equal2) show "0 < norm K - norm x" by (simp add: less_diff_eq 4) next fix h :: 'a assume "h ≠ 0" assume "norm (h - 0) < norm K - norm x" hence "norm x + norm h < norm K" by simp hence 5: "norm (x + h) < norm K" by (rule norm_triangle_ineq [THEN order_le_less_trans]) have A: "summable (λn. c n * x ^ n)" by (rule powser_inside [OF 1 4]) have B: "summable (λn. c n * (x + h) ^ n)" by (rule powser_inside [OF 1 5]) have C: "summable (λn. diffs c n * x ^ n)" by (rule powser_inside [OF 2 4]) show "((∑n. c n * (x + h) ^ n) - (∑n. c n * x ^ n)) / h - (∑n. diffs c n * x ^ n) = (∑n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))" apply (subst sums_unique [OF diffs_equiv [OF C]]) apply (subst suminf_diff [OF B A]) apply (subst suminf_divide [symmetric]) apply (rule summable_diff [OF B A]) apply (subst suminf_diff) apply (rule summable_divide) apply (rule summable_diff [OF B A]) apply (rule sums_summable [OF diffs_equiv [OF C]]) apply (rule_tac f="suminf" in arg_cong) apply (rule ext) apply (simp add: ring_simps) done next show "(λh. ∑n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" by (rule termdiffs_aux [OF 3 4]) qed qed subsection{*Exponential Function*} definition exp :: "'a => 'a::{recpower,real_normed_field,banach}" where "exp x = (∑n. x ^ n /R real (fact n))" definition sin :: "real => real" where "sin x = (∑n. (if even(n) then 0 else (-1 ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)" definition cos :: "real => real" where "cos x = (∑n. (if even(n) then (-1 ^ (n div 2))/(real (fact n)) else 0) * x ^ n)" lemma summable_exp_generic: fixes x :: "'a::{real_normed_algebra_1,recpower,banach}" defines S_def: "S ≡ λn. x ^ n /R real (fact n)" shows "summable S" proof - have S_Suc: "!!n. S (Suc n) = (x * S n) /R real (Suc n)" unfolding S_def by (simp add: power_Suc del: mult_Suc) obtain r :: real where r0: "0 < r" and r1: "r < 1" using dense [OF zero_less_one] by fast obtain N :: nat where N: "norm x < real N * r" using reals_Archimedean3 [OF r0] by fast from r1 show ?thesis proof (rule ratio_test [rule_format]) fix n :: nat assume n: "N ≤ n" have "norm x ≤ real N * r" using N by (rule order_less_imp_le) also have "real N * r ≤ real (Suc n) * r" using r0 n by (simp add: mult_right_mono) finally have "norm x * norm (S n) ≤ real (Suc n) * r * norm (S n)" using norm_ge_zero by (rule mult_right_mono) hence "norm (x * S n) ≤ real (Suc n) * r * norm (S n)" by (rule order_trans [OF norm_mult_ineq]) hence "norm (x * S n) / real (Suc n) ≤ r * norm (S n)" by (simp add: pos_divide_le_eq mult_ac) thus "norm (S (Suc n)) ≤ r * norm (S n)" by (simp add: S_Suc norm_scaleR inverse_eq_divide) qed qed lemma summable_norm_exp: fixes x :: "'a::{real_normed_algebra_1,recpower,banach}" shows "summable (λn. norm (x ^ n /R real (fact n)))" proof (rule summable_norm_comparison_test [OF exI, rule_format]) show "summable (λn. norm x ^ n /R real (fact n))" by (rule summable_exp_generic) next fix n show "norm (x ^ n /R real (fact n)) ≤ norm x ^ n /R real (fact n)" by (simp add: norm_scaleR norm_power_ineq) qed lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)" by (insert summable_exp_generic [where x=x], simp) lemma summable_sin: "summable (%n. (if even n then 0 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n)" apply (rule_tac g = "(%n. inverse (real (fact n)) * ¦x¦ ^ n)" in summable_comparison_test) apply (rule_tac [2] summable_exp) apply (rule_tac x = 0 in exI) apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) done lemma summable_cos: "summable (%n. (if even n then -1 ^ (n div 2)/(real (fact n)) else 0) * x ^ n)" apply (rule_tac g = "(%n. inverse (real (fact n)) * ¦x¦ ^ n)" in summable_comparison_test) apply (rule_tac [2] summable_exp) apply (rule_tac x = 0 in exI) apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) done lemma lemma_STAR_sin: "(if even n then 0 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0" by (induct "n", auto) lemma lemma_STAR_cos: "0 < n --> -1 ^ (n div 2)/(real (fact n)) * 0 ^ n = 0" by (induct "n", auto) lemma lemma_STAR_cos1: "0 < n --> (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0" by (induct "n", auto) lemma lemma_STAR_cos2: "(∑n=1..<n. if even n then -1 ^ (n div 2)/(real (fact n)) * 0 ^ n else 0) = 0" apply (induct "n") apply (case_tac [2] "n", auto) done lemma exp_converges: "(λn. x ^ n /R real (fact n)) sums exp x" unfolding exp_def by (rule summable_exp_generic [THEN summable_sums]) lemma sin_converges: "(%n. (if even n then 0 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n) sums sin(x)" unfolding sin_def by (rule summable_sin [THEN summable_sums]) lemma cos_converges: "(%n. (if even n then -1 ^ (n div 2)/(real (fact n)) else 0) * x ^ n) sums cos(x)" unfolding cos_def by (rule summable_cos [THEN summable_sums]) subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} lemma exp_fdiffs: "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))" by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult del: mult_Suc of_nat_Suc) lemma diffs_of_real: "diffs (λn. of_real (f n)) = (λn. of_real (diffs f n))" by (simp add: diffs_def) lemma sin_fdiffs: "diffs(%n. if even n then 0 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) = (%n. if even n then -1 ^ (n div 2)/(real (fact n)) else 0)" by (auto intro!: ext simp add: diffs_def divide_inverse real_of_nat_def of_nat_mult simp del: mult_Suc of_nat_Suc) lemma sin_fdiffs2: "diffs(%n. if even n then 0 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) n = (if even n then -1 ^ (n div 2)/(real (fact n)) else 0)" by (simp only: sin_fdiffs) lemma cos_fdiffs: "diffs(%n. if even n then -1 ^ (n div 2)/(real (fact n)) else 0) = (%n. - (if even n then 0 else -1 ^ ((n - Suc 0)div 2)/(real (fact n))))" by (auto intro!: ext simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def of_nat_mult simp del: mult_Suc of_nat_Suc) lemma cos_fdiffs2: "diffs(%n. if even n then -1 ^ (n div 2)/(real (fact n)) else 0) n = - (if even n then 0 else -1 ^ ((n - Suc 0)div 2)/(real (fact n)))" by (simp only: cos_fdiffs) text{*Now at last we can get the derivatives of exp, sin and cos*} lemma lemma_sin_minus: "- sin x = (∑n. - ((if even n then 0 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))" by (auto intro!: sums_unique sums_minus sin_converges) lemma lemma_exp_ext: "exp = (λx. ∑n. x ^ n /R real (fact n))" by (auto intro!: ext simp add: exp_def) lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)" apply (simp add: exp_def) apply (subst lemma_exp_ext) apply (subgoal_tac "DERIV (λu. ∑n. of_real (inverse (real (fact n))) * u ^ n) x :> (∑n. diffs (λn. of_real (inverse (real (fact n)))) n * x ^ n)") apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs) apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs) apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+ apply (simp del: of_real_add) done lemma lemma_sin_ext: "sin = (%x. ∑n. (if even n then 0 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n)" by (auto intro!: ext simp add: sin_def) lemma lemma_cos_ext: "cos = (%x. ∑n. (if even n then -1 ^ (n div 2)/(real (fact n)) else 0) * x ^ n)" by (auto intro!: ext simp add: cos_def) lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)" apply (simp add: cos_def) apply (subst lemma_sin_ext) apply (auto simp add: sin_fdiffs2 [symmetric]) apply (rule_tac K = "1 + ¦x¦" in termdiffs) apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs) done lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)" apply (subst lemma_cos_ext) apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left) apply (rule_tac K = "1 + ¦x¦" in termdiffs) apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus) done lemma isCont_exp [simp]: "isCont exp x" by (rule DERIV_exp [THEN DERIV_isCont]) lemma isCont_sin [simp]: "isCont sin x" by (rule DERIV_sin [THEN DERIV_isCont]) lemma isCont_cos [simp]: "isCont cos x" by (rule DERIV_cos [THEN DERIV_isCont]) subsection{*Properties of the Exponential Function*} lemma powser_zero: fixes f :: "nat => 'a::{real_normed_algebra_1,recpower}" shows "(∑n. f n * 0 ^ n) = f 0" proof - have "(∑n = 0..<1. f n * 0 ^ n) = (∑n. f n * 0 ^ n)" by (rule sums_unique [OF series_zero], simp add: power_0_left) thus ?thesis by simp qed lemma exp_zero [simp]: "exp 0 = 1" unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero) lemma setsum_head2: "m ≤ n ==> setsum f {m..n} = f m + setsum f {Suc m..n}" by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost) lemma setsum_cl_ivl_Suc2: "(∑i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (∑i=m..n. f (Suc i)))" by (simp add: setsum_head2 setsum_shift_bounds_cl_Suc_ivl del: setsum_cl_ivl_Suc) lemma exp_series_add: fixes x y :: "'a::{real_field,recpower}" defines S_def: "S ≡ λx n. x ^ n /R real (fact n)" shows "S (x + y) n = (∑i=0..n. S x i * S y (n - i))" proof (induct n) case 0 show ?case unfolding S_def by simp next case (Suc n) have S_Suc: "!!x n. S x (Suc n) = (x * S x n) /R real (Suc n)" unfolding S_def by (simp add: power_Suc del: mult_Suc) hence times_S: "!!x n. x * S x n = real (Suc n) *R S x (Suc n)" by simp have "real (Suc n) *R S (x + y) (Suc n) = (x + y) * S (x + y) n" by (simp only: times_S) also have "… = (x + y) * (∑i=0..n. S x i * S y (n-i))" by (simp only: Suc) also have "… = x * (∑i=0..n. S x i * S y (n-i)) + y * (∑i=0..n. S x i * S y (n-i))" by (rule left_distrib) also have "… = (∑i=0..n. (x * S x i) * S y (n-i)) + (∑i=0..n. S x i * (y * S y (n-i)))" by (simp only: setsum_right_distrib mult_ac) also have "… = (∑i=0..n. real (Suc i) *R (S x (Suc i) * S y (n-i))) + (∑i=0..n. real (Suc n-i) *R (S x i * S y (Suc n-i)))" by (simp add: times_S Suc_diff_le) also have "(∑i=0..n. real (Suc i) *R (S x (Suc i) * S y (n-i))) = (∑i=0..Suc n. real i *R (S x i * S y (Suc n-i)))" by (subst setsum_cl_ivl_Suc2, simp) also have "(∑i=0..n. real (Suc n-i) *R (S x i * S y (Suc n-i))) = (∑i=0..Suc n. real (Suc n-i) *R (S x i * S y (Suc n-i)))" by (subst setsum_cl_ivl_Suc, simp) also have "(∑i=0..Suc n. real i *R (S x i * S y (Suc n-i))) + (∑i=0..Suc n. real (Suc n-i) *R (S x i * S y (Suc n-i))) = (∑i=0..Suc n. real (Suc n) *R (S x i * S y (Suc n-i)))" by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric] real_of_nat_add [symmetric], simp) also have "… = real (Suc n) *R (∑i=0..Suc n. S x i * S y (Suc n-i))" by (simp only: scaleR_right.setsum) finally show "S (x + y) (Suc n) = (∑i=0..Suc n. S x i * S y (Suc n - i))" by (simp add: scaleR_cancel_left del: setsum_cl_ivl_Suc) qed lemma exp_add: "exp (x + y) = exp x * exp y" unfolding exp_def by (simp only: Cauchy_product summable_norm_exp exp_series_add) lemma exp_of_real: "exp (of_real x) = of_real (exp x)" unfolding exp_def apply (subst of_real.suminf) apply (rule summable_exp_generic) apply (simp add: scaleR_conv_of_real) done lemma exp_ge_add_one_self_aux: "0 ≤ (x::real) ==> (1 + x) ≤ exp(x)" apply (drule order_le_imp_less_or_eq, auto) apply (simp add: exp_def) apply (rule real_le_trans) apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le) apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff) done lemma exp_gt_one [simp]: "0 < (x::real) ==> 1 < exp x" apply (rule order_less_le_trans) apply (rule_tac [2] exp_ge_add_one_self_aux, auto) done lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)" proof - have "DERIV (exp o (λx. x + y)) x :> exp (x + y) * (1+0)" by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_ident DERIV_const) thus ?thesis by (simp add: o_def) qed lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)" proof - have "DERIV (exp o uminus) x :> exp (- x) * - 1" by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_ident) thus ?thesis by (simp add: o_def) qed lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0" proof - have "DERIV (λx. exp (x + y) * exp (- x)) x :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)" by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) thus ?thesis by (simp add: mult_commute) qed lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y::real)" proof - have "∀x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" by (rule DERIV_isconst_all) thus ?thesis by simp qed lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1" by (simp add: exp_add [symmetric]) lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1" by (simp add: mult_commute) lemma exp_minus: "exp(-x) = inverse(exp(x))" by (auto intro: inverse_unique [symmetric]) text{*Proof: because every exponential can be seen as a square.*} lemma exp_ge_zero [simp]: "0 ≤ exp (x::real)" apply (rule_tac t = x in real_sum_of_halves [THEN subst]) apply (subst exp_add, auto) done lemma exp_not_eq_zero [simp]: "exp x ≠ 0" apply (cut_tac x = x in exp_mult_minus2) apply (auto simp del: exp_mult_minus2) done lemma exp_gt_zero [simp]: "0 < exp (x::real)" by (simp add: order_less_le) lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x::real)" by (auto intro: positive_imp_inverse_positive) lemma abs_exp_cancel [simp]: "¦exp x::real¦ = exp x" by auto lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n" apply (induct "n") apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute) done lemma exp_diff: "exp(x - y) = exp(x)/(exp y)" apply (simp add: diff_minus divide_inverse) apply (simp (no_asm) add: exp_add exp_minus) done lemma exp_less_mono: fixes x y :: real assumes xy: "x < y" shows "exp x < exp y" proof - from xy have "1 < exp (y + - x)" by (rule real_less_sum_gt_zero [THEN exp_gt_one]) hence "exp x * inverse (exp x) < exp y * inverse (exp x)" by (auto simp add: exp_add exp_minus) thus ?thesis by (simp add: divide_inverse [symmetric] pos_less_divide_eq del: divide_self_if) qed lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y" apply (simp add: linorder_not_le [symmetric]) apply (auto simp add: order_le_less exp_less_mono) done lemma exp_less_cancel_iff [iff]: "(exp(x::real) < exp(y)) = (x < y)" by (auto intro: exp_less_mono exp_less_cancel) lemma exp_le_cancel_iff [iff]: "(exp(x::real) ≤ exp(y)) = (x ≤ y)" by (auto simp add: linorder_not_less [symmetric]) lemma exp_inj_iff [iff]: "(exp (x::real) = exp y) = (x = y)" by (simp add: order_eq_iff) lemma lemma_exp_total: "1 ≤ y ==> ∃x. 0 ≤ x & x ≤ y - 1 & exp(x::real) = y" apply (rule IVT) apply (auto intro: isCont_exp simp add: le_diff_eq) apply (subgoal_tac "1 + (y - 1) ≤ exp (y - 1)") apply simp apply (rule exp_ge_add_one_self_aux, simp) done lemma exp_total: "0 < (y::real) ==> ∃x. exp x = y" apply (rule_tac x = 1 and y = y in linorder_cases) apply (drule order_less_imp_le [THEN lemma_exp_total]) apply (rule_tac [2] x = 0 in exI) apply (frule_tac [3] real_inverse_gt_one) apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto) apply (rule_tac x = "-x" in exI) apply (simp add: exp_minus) done subsection{*Properties of the Logarithmic Function*} definition ln :: "real => real" where "ln x = (THE u. exp u = x)" lemma ln_exp [simp]: "ln (exp x) = x" by (simp add: ln_def) lemma exp_ln [simp]: "0 < x ==> exp (ln x) = x" by (auto dest: exp_total) lemma exp_ln_iff [simp]: "(exp (ln x) = x) = (0 < x)" apply (auto dest: exp_total) apply (erule subst, simp) done lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)" apply (rule exp_inj_iff [THEN iffD1]) apply (simp add: exp_add exp_ln mult_pos_pos) done lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)" apply (simp only: exp_ln_iff [symmetric]) apply (erule subst)+ apply simp done lemma ln_one[simp]: "ln 1 = 0" by (rule exp_inj_iff [THEN iffD1], auto) lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x" apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1]) apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric]) done lemma ln_div: "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y" apply (simp add: divide_inverse) apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse) done lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)" apply (simp only: exp_ln_iff [symmetric]) apply (erule subst)+ apply simp done lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x ≤ ln y) = (x ≤ y)" by (auto simp add: linorder_not_less [symmetric]) lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)" by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric]) lemma ln_add_one_self_le_self [simp]: "0 ≤ x ==> ln(1 + x) ≤ x" apply (rule ln_exp [THEN subst]) apply (rule ln_le_cancel_iff [THEN iffD2]) apply (auto simp add: exp_ge_add_one_self_aux) done lemma ln_less_self [simp]: "0 < x ==> ln x < x" apply (rule order_less_le_trans) apply (rule_tac [2] ln_add_one_self_le_self) apply (rule ln_less_cancel_iff [THEN iffD2], auto) done lemma ln_ge_zero [simp]: assumes x: "1 ≤ x" shows "0 ≤ ln x" proof - have "0 < x" using x by arith hence "exp 0 ≤ exp (ln x)" by (simp add: x) thus ?thesis by (simp only: exp_le_cancel_iff) qed lemma ln_ge_zero_imp_ge_one: assumes ln: "0 ≤ ln x" and x: "0 < x" shows "1 ≤ x" proof - from ln have "ln 1 ≤ ln x" by simp thus ?thesis by (simp add: x del: ln_one) qed lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 ≤ ln x) = (1 ≤ x)" by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one) lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)" by (insert ln_ge_zero_iff [of x], arith) lemma ln_gt_zero: assumes x: "1 < x" shows "0 < ln x" proof - have "0 < x" using x by arith hence "exp 0 < exp (ln x)" by (simp add: x) thus ?thesis by (simp only: exp_less_cancel_iff) qed lemma ln_gt_zero_imp_gt_one: assumes ln: "0 < ln x" and x: "0 < x" shows "1 < x" proof - from ln have "ln 1 < ln x" by simp thus ?thesis by (simp add: x del: ln_one) qed lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)" by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one) lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)" by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith) lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0" by simp lemma exp_ln_eq: "exp u = x ==> ln x = u" by auto lemma isCont_ln: "0 < x ==> isCont ln x" apply (subgoal_tac "isCont ln (exp (ln x))", simp) apply (rule isCont_inverse_function [where f=exp], simp_all) done lemma DERIV_ln: "0 < x ==> DERIV ln x :> inverse x" apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"]) apply (erule lemma_DERIV_subst [OF DERIV_exp exp_ln]) apply (simp_all add: abs_if isCont_ln) done subsection{*Basic Properties of the Trigonometric Functions*} lemma sin_zero [simp]: "sin 0 = 0" unfolding sin_def by (simp add: powser_zero) lemma cos_zero [simp]: "cos 0 = 1" unfolding cos_def by (simp add: powser_zero) lemma DERIV_sin_sin_mult [simp]: "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)" by (rule DERIV_mult, auto) lemma DERIV_sin_sin_mult2 [simp]: "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)" apply (cut_tac x = x in DERIV_sin_sin_mult) apply (auto simp add: mult_assoc) done lemma DERIV_sin_realpow2 [simp]: "DERIV (%x. (sin x)²) x :> cos(x) * sin(x) + cos(x) * sin(x)" by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric]) lemma DERIV_sin_realpow2a [simp]: "DERIV (%x. (sin x)²) x :> 2 * cos(x) * sin(x)" by (auto simp add: numeral_2_eq_2) lemma DERIV_cos_cos_mult [simp]: "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)" by (rule DERIV_mult, auto) lemma DERIV_cos_cos_mult2 [simp]: "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)" apply (cut_tac x = x in DERIV_cos_cos_mult) apply (auto simp add: mult_ac) done lemma DERIV_cos_realpow2 [simp]: "DERIV (%x. (cos x)²) x :> -sin(x) * cos(x) + -sin(x) * cos(x)" by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric]) lemma DERIV_cos_realpow2a [simp]: "DERIV (%x. (cos x)²) x :> -2 * cos(x) * sin(x)" by (auto simp add: numeral_2_eq_2) lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E" by auto lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)²) x :> -(2 * cos(x) * sin(x))" apply (rule lemma_DERIV_subst) apply (rule DERIV_cos_realpow2a, auto) done (* most useful *) lemma DERIV_cos_cos_mult3 [simp]: "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))" apply (rule lemma_DERIV_subst) apply (rule DERIV_cos_cos_mult2, auto) done lemma DERIV_sin_circle_all: "∀x. DERIV (%x. (sin x)² + (cos x)²) x :> (2*cos(x)*sin(x) - 2*cos(x)*sin(x))" apply (simp only: diff_minus, safe) apply (rule DERIV_add) apply (auto simp add: numeral_2_eq_2) done lemma DERIV_sin_circle_all_zero [simp]: "∀x. DERIV (%x. (sin x)² + (cos x)²) x :> 0" by (cut_tac DERIV_sin_circle_all, auto) lemma sin_cos_squared_add [simp]: "((sin x)²) + ((cos x)²) = 1" apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all]) apply (auto simp add: numeral_2_eq_2) done lemma sin_cos_squared_add2 [simp]: "((cos x)²) + ((sin x)²) = 1" apply (subst add_commute) apply (simp (no_asm) del: realpow_Suc) done lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1" apply (cut_tac x = x in sin_cos_squared_add2) apply (auto simp add: numeral_2_eq_2) done lemma sin_squared_eq: "(sin x)² = 1 - (cos x)²" apply (rule_tac a1 = "(cos x)²" in add_right_cancel [THEN iffD1]) apply (simp del: realpow_Suc) done lemma cos_squared_eq: "(cos x)² = 1 - (sin x)²" apply (rule_tac a1 = "(sin x)²" in add_right_cancel [THEN iffD1]) apply (simp del: realpow_Suc) done lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 ≤ y |] ==> 1 < x + (y::real)" by arith lemma abs_sin_le_one [simp]: "¦sin x¦ ≤ 1" by (rule power2_le_imp_le, simp_all add: sin_squared_eq) lemma sin_ge_minus_one [simp]: "-1 ≤ sin x" apply (insert abs_sin_le_one [of x]) apply (simp add: abs_le_iff del: abs_sin_le_one) done lemma sin_le_one [simp]: "sin x ≤ 1" apply (insert abs_sin_le_one [of x]) apply (simp add: abs_le_iff del: abs_sin_le_one) done lemma abs_cos_le_one [simp]: "¦cos x¦ ≤ 1" by (rule power2_le_imp_le, simp_all add: cos_squared_eq) lemma cos_ge_minus_one [simp]: "-1 ≤ cos x" apply (insert abs_cos_le_one [of x]) apply (simp add: abs_le_iff del: abs_cos_le_one) done lemma cos_le_one [simp]: "cos x ≤ 1" apply (insert abs_cos_le_one [of x]) apply (simp add: abs_le_iff del: abs_cos_le_one) done lemma DERIV_fun_pow: "DERIV g x :> m ==> DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m" apply (rule lemma_DERIV_subst) apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2) apply (rule DERIV_pow, auto) done lemma DERIV_fun_exp: "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m" apply (rule lemma_DERIV_subst) apply (rule_tac f = exp in DERIV_chain2) apply (rule DERIV_exp, auto) done lemma DERIV_fun_sin: "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m" apply (rule lemma_DERIV_subst) apply (rule_tac f = sin in DERIV_chain2) apply (rule DERIV_sin, auto) done lemma DERIV_fun_cos: "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m" apply (rule lemma_DERIV_subst) apply (rule_tac f = cos in DERIV_chain2) apply (rule DERIV_cos, auto) done lemmas DERIV_intros = DERIV_ident DERIV_const DERIV_cos DERIV_cmult DERIV_sin DERIV_exp DERIV_inverse DERIV_pow DERIV_add DERIV_diff DERIV_mult DERIV_minus DERIV_inverse_fun DERIV_quotient DERIV_fun_pow DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos (* lemma *) lemma lemma_DERIV_sin_cos_add: "∀x. DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 + (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0" apply (safe, rule lemma_DERIV_subst) apply (best intro!: DERIV_intros intro: DERIV_chain2) --{*replaces the old @{text DERIV_tac}*} apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac) done lemma sin_cos_add [simp]: "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 + (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0" apply (cut_tac y = 0 and x = x and y7 = y in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all]) apply (auto simp add: numeral_2_eq_2) done lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y" apply (cut_tac x = x and y = y in sin_cos_add) apply (simp del: sin_cos_add) done lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y" apply (cut_tac x = x and y = y in sin_cos_add) apply (simp del: sin_cos_add) done lemma lemma_DERIV_sin_cos_minus: "∀x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0" apply (safe, rule lemma_DERIV_subst) apply (best intro!: DERIV_intros intro: DERIV_chain2) apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac) done lemma sin_cos_minus [simp]: "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0" apply (cut_tac y = 0 and x = x in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all]) apply simp done lemma sin_minus [simp]: "sin (-x) = -sin(x)" apply (cut_tac x = x in sin_cos_minus) apply (simp del: sin_cos_minus) done lemma cos_minus [simp]: "cos (-x) = cos(x)" apply (cut_tac x = x in sin_cos_minus) apply (simp del: sin_cos_minus) done lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y" by (simp add: diff_minus sin_add) lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x" by (simp add: sin_diff mult_commute) lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y" by (simp add: diff_minus cos_add) lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x" by (simp add: cos_diff mult_commute) lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x" by (cut_tac x = x and y = x in sin_add, auto) lemma cos_double: "cos(2* x) = ((cos x)²) - ((sin x)²)" apply (cut_tac x = x and y = x in cos_add) apply (simp add: power2_eq_square) done subsection{*The Constant Pi*} definition pi :: "real" where "pi = 2 * (THE x. 0 ≤ (x::real) & x ≤ 2 & cos x = 0)" text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; hence define pi.*} lemma sin_paired: "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) sums sin x" proof - have "(λn. ∑k = n * 2..<n * 2 + 2. (if even k then 0 else -1 ^ ((k - Suc 0) div 2) / real (fact k)) * x ^ k) sums sin x" unfolding sin_def by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) thus ?thesis by (simp add: mult_ac) qed lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x" apply (subgoal_tac "(λn. ∑k = n * 2..<n * 2 + 2. -1 ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) sums (∑n. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))") prefer 2 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) apply (rotate_tac 2) apply (drule sin_paired [THEN sums_unique, THEN ssubst]) apply (auto simp del: fact_Suc realpow_Suc) apply (frule sums_unique) apply (auto simp del: fact_Suc realpow_Suc) apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans]) apply (auto simp del: fact_Suc realpow_Suc) apply (erule sums_summable) apply (case_tac "m=0") apply (simp (no_asm_simp)) apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x") apply (simp only: mult_less_cancel_left, simp) apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric]) apply (subgoal_tac "x*x < 2*3", simp) apply (rule mult_strict_mono) apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc) apply (subst fact_Suc) apply (subst fact_Suc) apply (subst fact_Suc) apply (subst fact_Suc) apply (subst real_of_nat_mult) apply (subst real_of_nat_mult) apply (subst real_of_nat_mult) apply (subst real_of_nat_mult) apply (simp (no_asm) add: divide_inverse del: fact_Suc) apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc) apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) apply (auto simp add: mult_assoc simp del: fact_Suc) apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc) apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") apply (erule ssubst)+ apply (auto simp del: fact_Suc) apply (subgoal_tac "0 < x ^ (4 * m) ") prefer 2 apply (simp only: zero_less_power) apply (simp (no_asm_simp) add: mult_less_cancel_left) apply (rule mult_strict_mono) apply (simp_all (no_asm_simp)) done lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x" by (auto intro: sin_gt_zero) lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1" apply (cut_tac x = x in sin_gt_zero1) apply (auto simp add: cos_squared_eq cos_double) done lemma cos_paired: "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x" proof - have "(λn. ∑k = n * 2..<n * 2 + 2. (if even k then -1 ^ (k div 2) / real (fact k) else 0) * x ^ k) sums cos x" unfolding cos_def by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) thus ?thesis by (simp add: mult_ac) qed declare zero_less_power [simp] lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)" by simp lemma cos_two_less_zero [simp]: "cos (2) < 0" apply (cut_tac x = 2 in cos_paired) apply (drule sums_minus) apply (rule neg_less_iff_less [THEN iffD1]) apply (frule sums_unique, auto) apply (rule_tac y = "∑n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))" in order_less_trans) apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc) apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc) apply (rule sumr_pos_lt_pair) apply (erule sums_summable, safe) apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] del: fact_Suc) apply (rule real_mult_inverse_cancel2) apply (rule real_of_nat_fact_gt_zero)+ apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc) apply (subst fact_lemma) apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"]) apply (simp only: real_of_nat_mult) apply (rule mult_strict_mono, force) apply (rule_tac [3] real_of_nat_fact_ge_zero) prefer 2 apply force apply (rule real_of_nat_less_iff [THEN iffD2]) apply (rule fact_less_mono, auto) done lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq] lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le] lemma cos_is_zero: "EX! x. 0 ≤ x & x ≤ 2 & cos x = 0" apply (subgoal_tac "∃x. 0 ≤ x & x ≤ 2 & cos x = 0") apply (rule_tac [2] IVT2) apply (auto intro: DERIV_isCont DERIV_cos) apply (cut_tac x = xa and y = y in linorder_less_linear) apply (rule ccontr) apply (subgoal_tac " (∀x. cos differentiable x) & (∀x. isCont cos x) ") apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def) apply (drule_tac f = cos in Rolle) apply (drule_tac [5] f = cos in Rolle) apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def) apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero]) apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) done lemma pi_half: "pi/2 = (THE x. 0 ≤ x & x ≤ 2 & cos x = 0)" by (simp add: pi_def) lemma cos_pi_half [simp]: "cos (pi / 2) = 0" by (simp add: pi_half cos_is_zero [THEN theI']) lemma pi_half_gt_zero [simp]: "0 < pi / 2" apply (rule order_le_neq_trans) apply (simp add: pi_half cos_is_zero [THEN theI']) apply (rule notI, drule arg_cong [where f=cos], simp) done lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric] lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le] lemma pi_half_less_two [simp]: "pi / 2 < 2" apply (rule order_le_neq_trans) apply (simp add: pi_half cos_is_zero [THEN theI']) apply (rule notI, drule arg_cong [where f=cos], simp) done lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq] lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le] lemma pi_gt_zero [simp]: "0 < pi" by (insert pi_half_gt_zero, simp) lemma pi_ge_zero [simp]: "0 ≤ pi" by (rule pi_gt_zero [THEN order_less_imp_le]) lemma pi_neq_zero [simp]: "pi ≠ 0" by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym]) lemma pi_not_less_zero [simp]: "¬ pi < 0" by (simp add: linorder_not_less) lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0" by auto lemma sin_pi_half [simp]: "sin(pi/2) = 1" apply (cut_tac x = "pi/2" in sin_cos_squared_add2) apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two]) apply (simp add: power2_eq_square) done lemma cos_pi [simp]: "cos pi = -1" by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp) lemma sin_pi [simp]: "sin pi = 0" by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp) lemma sin_cos_eq: "sin x = cos (pi/2 - x)" by (simp add: diff_minus cos_add) declare sin_cos_eq [symmetric, simp] lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)" by (simp add: cos_add) declare minus_sin_cos_eq [symmetric, simp] lemma cos_sin_eq: "cos x = sin (pi/2 - x)" by (simp add: diff_minus sin_add) declare cos_sin_eq [symmetric, simp] lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x" by (simp add: sin_add) lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x" by (simp add: sin_add) lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x" by (simp add: cos_add) lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x" by (simp add: sin_add cos_double) lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x" by (simp add: cos_add cos_double) lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n" apply (induct "n") apply (auto simp add: real_of_nat_Suc left_distrib) done lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n" proof - have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute) also have "... = -1 ^ n" by (rule cos_npi) finally show ?thesis . qed lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0" apply (induct "n") apply (auto simp add: real_of_nat_Suc left_distrib) done lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0" by (simp add: mult_commute [of pi]) lemma cos_two_pi [simp]: "cos (2 * pi) = 1" by (simp add: cos_double) lemma sin_two_pi [simp]: "sin (2 * pi) = 0" by simp lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x" apply (rule sin_gt_zero, assumption) apply (rule order_less_trans, assumption) apply (rule pi_half_less_two) done lemma sin_less_zero: assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0" proof - have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) thus ?thesis by simp qed lemma pi_less_4: "pi < 4" by (cut_tac pi_half_less_two, auto) lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x" apply (cut_tac pi_less_4) apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all) apply (cut_tac cos_is_zero, safe) apply (rename_tac y z) apply (drule_tac x = y in spec) apply (drule_tac x = "pi/2" in spec, simp) done lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x" apply (rule_tac x = x and y = 0 in linorder_cases) apply (rule cos_minus [THEN subst]) apply (rule cos_gt_zero) apply (auto intro: cos_gt_zero) done lemma cos_ge_zero: "[| -(pi/2) ≤ x; x ≤ pi/2 |] ==> 0 ≤ cos x" apply (auto simp add: order_le_less cos_gt_zero_pi) apply (subgoal_tac "x = pi/2", auto) done lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x" apply (subst sin_cos_eq) apply (rotate_tac 1) apply (drule real_sum_of_halves [THEN ssubst]) apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric]) done lemma sin_ge_zero: "[| 0 ≤ x; x ≤ pi |] ==> 0 ≤ sin x" by (auto simp add: order_le_less sin_gt_zero_pi) lemma cos_total: "[| -1 ≤ y; y ≤ 1 |] ==> EX! x. 0 ≤ x & x ≤ pi & (cos x = y)" apply (subgoal_tac "∃x. 0 ≤ x & x ≤ pi & cos x = y") apply (rule_tac [2] IVT2) apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos) apply (cut_tac x = xa and y = y in linorder_less_linear) apply (rule ccontr, auto) apply (drule_tac f = cos in Rolle) apply (drule_tac [5] f = cos in Rolle) apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def) apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans]) done lemma sin_total: "[| -1 ≤ y; y ≤ 1 |] ==> EX! x. -(pi/2) ≤ x & x ≤ pi/2 & (sin x = y)" apply (rule ccontr) apply (subgoal_tac "∀x. (- (pi/2) ≤ x & x ≤ pi/2 & (sin x = y)) = (0 ≤ (x + pi/2) & (x + pi/2) ≤ pi & (cos (x + pi/2) = -y))") apply (erule contrapos_np) apply (simp del: minus_sin_cos_eq [symmetric]) apply (cut_tac y="-y" in cos_total, simp) apply simp apply (erule ex1E) apply (rule_tac a = "x - (pi/2)" in ex1I) apply (simp (no_asm) add: add_assoc) apply (rotate_tac 3) apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) done lemma reals_Archimedean4: "[| 0 < y; 0 ≤ x |] ==> ∃n. real n * y ≤ x & x < real (Suc n) * y" apply (auto dest!: reals_Archimedean3) apply (drule_tac x = x in spec, clarify) apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y") prefer 2 apply (erule LeastI) apply (case_tac "LEAST m::nat. x < real m * y", simp) apply (subgoal_tac "~ x < real nat * y") prefer 2 apply (rule not_less_Least, simp, force) done (* Pre Isabelle99-2 proof was simpler- numerals arithmetic now causes some unwanted re-arrangements of literals! *) lemma cos_zero_lemma: "[| 0 ≤ x; cos x = 0 |] ==> ∃n::nat. ~even n & x = real n * (pi/2)" apply (drule pi_gt_zero [THEN reals_Archimedean4], safe) apply (subgoal_tac "0 ≤ x - real n * pi & (x - real n * pi) ≤ pi & (cos (x - real n * pi) = 0) ") apply (auto simp add: compare_rls) prefer 3 apply (simp add: cos_diff) prefer 2 apply (simp add: real_of_nat_Suc left_distrib) apply (simp add: cos_diff) apply (subgoal_tac "EX! x. 0 ≤ x & x ≤ pi & cos x = 0") apply (rule_tac [2] cos_total, safe) apply (drule_tac x = "x - real n * pi" in spec) apply (drule_tac x = "pi/2" in spec) apply (simp add: cos_diff) apply (rule_tac x = "Suc (2 * n)" in exI) apply (simp add: real_of_nat_Suc left_distrib, auto) done lemma sin_zero_lemma: "[| 0 ≤ x; sin x = 0 |] ==> ∃n::nat. even n & x = real n * (pi/2)" apply (subgoal_tac "∃n::nat. ~ even n & x + pi/2 = real n * (pi/2) ") apply (clarify, rule_tac x = "n - 1" in exI) apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) apply (rule cos_zero_lemma) apply (simp_all add: add_increasing) done lemma cos_zero_iff: "(cos x = 0) = ((∃n::nat. ~even n & (x = real n * (pi/2))) | (∃n::nat. ~even n & (x = -(real n * (pi/2)))))" apply (rule iffI) apply (cut_tac linorder_linear [of 0 x], safe) apply (drule cos_zero_lemma, assumption+) apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) apply (force simp add: minus_equation_iff [of x]) apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) apply (auto simp add: cos_add) done (* ditto: but to a lesser extent *) lemma sin_zero_iff: "(sin x = 0) = ((∃n::nat. even n & (x = real n * (pi/2))) | (∃n::nat. even n & (x = -(real n * (pi/2)))))" apply (rule iffI) apply (cut_tac linorder_linear [of 0 x], safe) apply (drule sin_zero_lemma, assumption+) apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe) apply (force simp add: minus_equation_iff [of x]) apply (auto simp add: even_mult_two_ex) done subsection{*Tangent*} definition tan :: "real => real" where "tan x = (sin x)/(cos x)" lemma tan_zero [simp]: "tan 0 = 0" by (simp add: tan_def) lemma tan_pi [simp]: "tan pi = 0" by (simp add: tan_def) lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0" by (simp add: tan_def) lemma tan_minus [simp]: "tan (-x) = - tan x" by (simp add: tan_def minus_mult_left) lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x" by (simp add: tan_def) lemma lemma_tan_add1: "[| cos x ≠ 0; cos y ≠ 0 |] ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)" apply (simp add: tan_def divide_inverse) apply (auto simp del: inverse_mult_distrib simp add: inverse_mult_distrib [symmetric] mult_ac) apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst]) apply (auto simp del: inverse_mult_distrib simp add: mult_assoc left_diff_distrib cos_add) done lemma add_tan_eq: "[| cos x ≠ 0; cos y ≠ 0 |] ==> tan x + tan y = sin(x + y)/(cos x * cos y)" apply (simp add: tan_def) apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst]) apply (auto simp add: mult_assoc left_distrib) apply (simp add: sin_add) done lemma tan_add: "[| cos x ≠ 0; cos y ≠ 0; cos (x + y) ≠ 0 |] ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))" apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1) apply (simp add: tan_def) done lemma tan_double: "[| cos x ≠ 0; cos (2 * x) ≠ 0 |] ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))" apply (insert tan_add [of x x]) apply (simp add: mult_2 [symmetric]) apply (auto simp add: numeral_2_eq_2) done lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x" by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) lemma tan_less_zero: assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0" proof - have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) thus ?thesis by simp qed lemma lemma_DERIV_tan: "cos x ≠ 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)²)" apply (rule lemma_DERIV_subst) apply (best intro!: DERIV_intros intro: DERIV_chain2) apply (auto simp add: divide_inverse numeral_2_eq_2) done lemma DERIV_tan [simp]: "cos x ≠ 0 ==> DERIV tan x :> inverse((cos x)²)" by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric]) lemma isCont_tan [simp]: "cos x ≠ 0 ==> isCont tan x" by (rule DERIV_tan [THEN DERIV_isCont]) lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0" apply (subgoal_tac "(λx. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1") apply (simp add: divide_inverse [symmetric]) apply (rule LIM_mult) apply (rule_tac [2] inverse_1 [THEN subst]) apply (rule_tac [2] LIM_inverse) apply (simp_all add: divide_inverse [symmetric]) apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+ done lemma lemma_tan_total: "0 < y ==> ∃x. 0 < x & x < pi/2 & y < tan x" apply (cut_tac LIM_cos_div_sin) apply (simp only: LIM_def) apply (drule_tac x = "inverse y" in spec, safe, force) apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe) apply (rule_tac x = "(pi/2) - e" in exI) apply (simp (no_asm_simp)) apply (drule_tac x = "(pi/2) - e" in spec) apply (auto simp add: tan_def) apply (rule inverse_less_iff_less [THEN iffD1]) apply (auto simp add: divide_inverse) apply (rule real_mult_order) apply (subgoal_tac [3] "0 < sin e & 0 < cos e") apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) done lemma tan_total_pos: "0 ≤ y ==> ∃x. 0 ≤ x & x < pi/2 & tan x = y" apply (frule order_le_imp_less_or_eq, safe) prefer 2 apply force apply (drule lemma_tan_total, safe) apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl) apply (auto intro!: DERIV_tan [THEN DERIV_isCont]) apply (drule_tac y = xa in order_le_imp_less_or_eq) apply (auto dest: cos_gt_zero) done lemma lemma_tan_total1: "∃x. -(pi/2) < x & x < (pi/2) & tan x = y" apply (cut_tac linorder_linear [of 0 y], safe) apply (drule tan_total_pos) apply (cut_tac [2] y="-y" in tan_total_pos, safe) apply (rule_tac [3] x = "-x" in exI) apply (auto intro!: exI) done lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y" apply (cut_tac y = y in lemma_tan_total1, auto) apply (cut_tac x = xa and y = y in linorder_less_linear, auto) apply (subgoal_tac [2] "∃z. y < z & z < xa & DERIV tan z :> 0") apply (subgoal_tac "∃z. xa < z & z < y & DERIV tan z :> 0") apply (rule_tac [4] Rolle) apply (rule_tac [2] Rolle) apply (auto intro!: DERIV_tan DERIV_isCont exI simp add: differentiable_def) txt{*Now, simulate TRYALL*} apply (rule_tac [!] DERIV_tan asm_rl) apply (auto dest!: DERIV_unique [OF _ DERIV_tan] simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym]) done subsection {* Inverse Trigonometric Functions *} definition arcsin :: "real => real" where "arcsin y = (THE x. -(pi/2) ≤ x & x ≤ pi/2 & sin x = y)" definition arccos :: "real => real" where "arccos y = (THE x. 0 ≤ x & x ≤ pi & cos x = y)" definition arctan :: "real => real" where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)" lemma arcsin: "[| -1 ≤ y; y ≤ 1 |] ==> -(pi/2) ≤ arcsin y & arcsin y ≤ pi/2 & sin(arcsin y) = y" unfolding arcsin_def by (rule theI' [OF sin_total]) lemma arcsin_pi: "[| -1 ≤ y; y ≤ 1 |] ==> -(pi/2) ≤ arcsin y & arcsin y ≤ pi & sin(arcsin y) = y" apply (drule (1) arcsin) apply (force intro: order_trans) done lemma sin_arcsin [simp]: "[| -1 ≤ y; y ≤ 1 |] ==> sin(arcsin y) = y" by (blast dest: arcsin) lemma arcsin_bounded: "[| -1 ≤ y; y ≤ 1 |] ==> -(pi/2) ≤ arcsin y & arcsin y ≤ pi/2" by (blast dest: arcsin) lemma arcsin_lbound: "[| -1 ≤ y; y ≤ 1 |] ==> -(pi/2) ≤ arcsin y" by (blast dest: arcsin) lemma arcsin_ubound: "[| -1 ≤ y; y ≤ 1 |] ==> arcsin y ≤ pi/2" by (blast dest: arcsin) lemma arcsin_lt_bounded: "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2" apply (frule order_less_imp_le) apply (frule_tac y = y in order_less_imp_le) apply (frule arcsin_bounded) apply (safe, simp) apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq) apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe) apply (drule_tac [!] f = sin in arg_cong, auto) done lemma arcsin_sin: "[|-(pi/2) ≤ x; x ≤ pi/2 |] ==> arcsin(sin x) = x" apply (unfold arcsin_def) apply (rule the1_equality) apply (rule sin_total, auto) done lemma arccos: "[| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arccos y & arccos y ≤ pi & cos(arccos y) = y" unfolding arccos_def by (rule theI' [OF cos_total]) lemma cos_arccos [simp]: "[| -1 ≤ y; y ≤ 1 |] ==> cos(arccos y) = y" by (blast dest: arccos) lemma arccos_bounded: "[| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arccos y & arccos y ≤ pi" by (blast dest: arccos) lemma arccos_lbound: "[| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arccos y" by (blast dest: arccos) lemma arccos_ubound: "[| -1 ≤ y; y ≤ 1 |] ==> arccos y ≤ pi" by (blast dest: arccos) lemma arccos_lt_bounded: "[| -1 < y; y < 1 |] ==> 0 < arccos y & arccos y < pi" apply (frule order_less_imp_le) apply (frule_tac y = y in order_less_imp_le) apply (frule arccos_bounded, auto) apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq) apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto) apply (drule_tac [!] f = cos in arg_cong, auto) done lemma arccos_cos: "[|0 ≤ x; x ≤ pi |] ==> arccos(cos x) = x" apply (simp add: arccos_def) apply (auto intro!: the1_equality cos_total) done lemma arccos_cos2: "[|x ≤ 0; -pi ≤ x |] ==> arccos(cos x) = -x" apply (simp add: arccos_def) apply (auto intro!: the1_equality cos_total) done lemma cos_arcsin: "[|-1 ≤ x; x ≤ 1|] ==> cos (arcsin x) = sqrt (1 - x²)" apply (subgoal_tac "x² ≤ 1") apply (rule power2_eq_imp_eq) apply (simp add: cos_squared_eq) apply (rule cos_ge_zero) apply (erule (1) arcsin_lbound) apply (erule (1) arcsin_ubound) apply simp apply (subgoal_tac "¦x¦² ≤ 1²", simp) apply (rule power_mono, simp, simp) done lemma sin_arccos: "[|-1 ≤ x; x ≤ 1|] ==> sin (arccos x) = sqrt (1 - x²)" apply (subgoal_tac "x² ≤ 1") apply (rule power2_eq_imp_eq) apply (simp add: sin_squared_eq) apply (rule sin_ge_zero) apply (erule (1) arccos_lbound) apply (erule (1) arccos_ubound) apply simp apply (subgoal_tac "¦x¦² ≤ 1²", simp) apply (rule power_mono, simp, simp) done lemma arctan [simp]: "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y" unfolding arctan_def by (rule theI' [OF tan_total]) lemma tan_arctan: "tan(arctan y) = y" by auto lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2" by (auto simp only: arctan) lemma arctan_lbound: "- (pi/2) < arctan y" by auto lemma arctan_ubound: "arctan y < pi/2" by (auto simp only: arctan) lemma arctan_tan: "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x" apply (unfold arctan_def) apply (rule the1_equality) apply (rule tan_total, auto) done lemma arctan_zero_zero [simp]: "arctan 0 = 0" by (insert arctan_tan [of 0], simp) lemma cos_arctan_not_zero [simp]: "cos(arctan x) ≠ 0" apply (auto simp add: cos_zero_iff) apply (case_tac "n") apply (case_tac [3] "n") apply (cut_tac [2] y = x in arctan_ubound) apply (cut_tac [4] y = x in arctan_lbound) apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff) done lemma tan_sec: "cos x ≠ 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2" apply (rule power_inverse [THEN subst]) apply (rule_tac c1 = "(cos x)²" in real_mult_right_cancel [THEN iffD1]) apply (auto dest: field_power_not_zero simp add: power_mult_distrib left_distrib power_divide tan_def mult_assoc power_inverse [symmetric] simp del: realpow_Suc) done lemma isCont_inverse_function2: fixes f g :: "real => real" shows "[|a < x; x < b; ∀z. a ≤ z ∧ z ≤ b --> g (f z) = z; ∀z. a ≤ z ∧ z ≤ b --> isCont f z|] ==> isCont g (f x)" apply (rule isCont_inverse_function [where f=f and d="min (x - a) (b - x)"]) apply (simp_all add: abs_le_iff) done lemma isCont_arcsin: "[|-1 < x; x < 1|] ==> isCont arcsin x" apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp) apply (rule isCont_inverse_function2 [where f=sin]) apply (erule (1) arcsin_lt_bounded [THEN conjunct1]) apply (erule (1) arcsin_lt_bounded [THEN conjunct2]) apply (fast intro: arcsin_sin, simp) done lemma isCont_arccos: "[|-1 < x; x < 1|] ==> isCont arccos x" apply (subgoal_tac "isCont arccos (cos (arccos x))", simp) apply (rule isCont_inverse_function2 [where f=cos]) apply (erule (1) arccos_lt_bounded [THEN conjunct1]) apply (erule (1) arccos_lt_bounded [THEN conjunct2]) apply (fast intro: arccos_cos, simp) done lemma isCont_arctan: "isCont arctan x" apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify) apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify) apply (subgoal_tac "isCont arctan (tan (arctan x))", simp) apply (erule (1) isCont_inverse_function2 [where f=tan]) apply (clarify, rule arctan_tan) apply (erule (1) order_less_le_trans) apply (erule (1) order_le_less_trans) apply (clarify, rule isCont_tan) apply (rule less_imp_neq [symmetric]) apply (rule cos_gt_zero_pi) apply (erule (1) order_less_le_trans) apply (erule (1) order_le_less_trans) done lemma DERIV_arcsin: "[|-1 < x; x < 1|] ==> DERIV arcsin x :> inverse (sqrt (1 - x²))" apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"]) apply (rule lemma_DERIV_subst [OF DERIV_sin]) apply (simp add: cos_arcsin) apply (subgoal_tac "¦x¦² < 1²", simp) apply (rule power_strict_mono, simp, simp, simp) apply assumption apply assumption apply simp apply (erule (1) isCont_arcsin) done lemma DERIV_arccos: "[|-1 < x; x < 1|] ==> DERIV arccos x :> inverse (- sqrt (1 - x²))" apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"]) apply (rule lemma_DERIV_subst [OF DERIV_cos]) apply (simp add: sin_arccos) apply (subgoal_tac "¦x¦² < 1²", simp) apply (rule power_strict_mono, simp, simp, simp) apply assumption apply assumption apply simp apply (erule (1) isCont_arccos) done lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x²)" apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"]) apply (rule lemma_DERIV_subst [OF DERIV_tan]) apply (rule cos_arctan_not_zero) apply (simp add: power_inverse tan_sec [symmetric]) apply (subgoal_tac "0 < 1 + x²", simp) apply (simp add: add_pos_nonneg) apply (simp, simp, simp, rule isCont_arctan) done subsection {* More Theorems about Sin and Cos *} lemma cos_45: "cos (pi / 4) = sqrt 2 / 2" proof - let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)" have nonneg: "0 ≤ ?c" by (rule cos_ge_zero, rule order_trans [where y=0], simp_all) have "0 = cos (pi / 4 + pi / 4)" by simp also have "cos (pi / 4 + pi / 4) = ?c² - ?s²" by (simp only: cos_add power2_eq_square) also have "… = 2 * ?c² - 1" by (simp add: sin_squared_eq) finally have "?c² = (sqrt 2 / 2)²" by (simp add: power_divide) thus ?thesis using nonneg by (rule power2_eq_imp_eq) simp qed lemma cos_30: "cos (pi / 6) = sqrt 3 / 2" proof - let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)" have pos_c: "0 < ?c" by (rule cos_gt_zero, simp, simp) have "0 = cos (pi / 6 + pi / 6 + pi / 6)" by simp also have "… = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s" by (simp only: cos_add sin_add) also have "… = ?c * (?c² - 3 * ?s²)" by (simp add: ring_simps power2_eq_square) finally have "?c² = (sqrt 3 / 2)²" using pos_c by (simp add: sin_squared_eq power_divide) thus ?thesis using pos_c [THEN order_less_imp_le] by (rule power2_eq_imp_eq) simp qed lemma sin_45: "sin (pi / 4) = sqrt 2 / 2" proof - have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq) also have "pi / 2 - pi / 4 = pi / 4" by simp also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45) finally show ?thesis . qed lemma sin_60: "sin (pi / 3) = sqrt 3 / 2" proof - have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq) also have "pi / 2 - pi / 3 = pi / 6" by simp also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30) finally show ?thesis . qed lemma cos_60: "cos (pi / 3) = 1 / 2" apply (rule power2_eq_imp_eq) apply (simp add: cos_squared_eq sin_60 power_divide) apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all) done lemma sin_30: "sin (pi / 6) = 1 / 2" proof - have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq) also have "pi / 2 - pi / 6 = pi / 3" by simp also have "cos (pi / 3) = 1 / 2" by (rule cos_60) finally show ?thesis . qed lemma tan_30: "tan (pi / 6) = 1 / sqrt 3" unfolding tan_def by (simp add: sin_30 cos_30) lemma tan_45: "tan (pi / 4) = 1" unfolding tan_def by (simp add: sin_45 cos_45) lemma tan_60: "tan (pi / 3) = sqrt 3" unfolding tan_def by (simp add: sin_60 cos_60) text{*NEEDED??*} lemma [simp]: "sin (x + 1 / 2 * real (Suc m) * pi) = cos (x + 1 / 2 * real (m) * pi)" by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto) text{*NEEDED??*} lemma [simp]: "sin (x + real (Suc m) * pi / 2) = cos (x + real (m) * pi / 2)" by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto) lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)" apply (rule lemma_DERIV_subst) apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2) apply (best intro!: DERIV_intros intro: DERIV_chain2)+ apply (simp (no_asm)) done lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n" proof - have "sin ((real n + 1/2) * pi) = cos (real n * pi)" by (auto simp add: right_distrib sin_add left_distrib mult_ac) thus ?thesis by (simp add: real_of_nat_Suc left_distrib add_divide_distrib mult_commute [of pi]) qed lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1" by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2) lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0" apply (subgoal_tac "cos (pi + pi/2) = 0", simp) apply (subst cos_add, simp) done lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0" by (auto simp add: mult_assoc) lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1" apply (subgoal_tac "sin (pi + pi/2) = - 1", simp) apply (subst sin_add, simp) done (*NEEDED??*) lemma [simp]: "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)" apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto) done (*NEEDED??*) lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)" by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto) lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0" by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto) lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)" apply (rule lemma_DERIV_subst) apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2) apply (best intro!: DERIV_intros intro: DERIV_chain2)+ apply (simp (no_asm)) done lemma sin_zero_abs_cos_one: "sin x = 0 ==> ¦cos x¦ = 1" by (auto simp add: sin_zero_iff even_mult_two_ex) lemma exp_eq_one_iff [simp]: "(exp (x::real) = 1) = (x = 0)" apply auto apply (drule_tac f = ln in arg_cong, auto) done lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0" by (cut_tac x = x in sin_cos_squared_add3, auto) subsection {* Existence of Polar Coordinates *} lemma cos_x_y_le_one: "¦x / sqrt (x² + y²)¦ ≤ 1" apply (rule power2_le_imp_le [OF _ zero_le_one]) apply (simp add: abs_divide power_divide divide_le_eq not_sum_power2_lt_zero) done lemma cos_arccos_abs: "¦y¦ ≤ 1 ==> cos (arccos y) = y" by (simp add: abs_le_iff) lemma sin_arccos_abs: "¦y¦ ≤ 1 ==> sin (arccos y) = sqrt (1 - y²)" by (simp add: sin_arccos abs_le_iff) lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one] lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one] lemma polar_ex1: "0 < y ==> ∃r a. x = r * cos a & y = r * sin a" apply (rule_tac x = "sqrt (x² + y²)" in exI) apply (rule_tac x = "arccos (x / sqrt (x² + y²))" in exI) apply (simp add: cos_arccos_lemma1) apply (simp add: sin_arccos_lemma1) apply (simp add: power_divide) apply (simp add: real_sqrt_mult [symmetric]) apply (simp add: right_diff_distrib) done lemma polar_ex2: "y < 0 ==> ∃r a. x = r * cos a & y = r * sin a" apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify) apply (rule_tac x = r in exI) apply (rule_tac x = "-a" in exI, simp) done lemma polar_Ex: "∃r a. x = r * cos a & y = r * sin a" apply (rule_tac x=0 and y=y in linorder_cases) apply (erule polar_ex1) apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp) apply (erule polar_ex2) done subsection {* Theorems about Limits *} (* need to rename second isCont_inverse *) lemma isCont_inv_fun: fixes f g :: "real => real" shows "[| 0 < d; ∀z. ¦z - x¦ ≤ d --> g(f(z)) = z; ∀z. ¦z - x¦ ≤ d --> isCont f z |] ==> isCont g (f x)" by (rule isCont_inverse_function) lemma isCont_inv_fun_inv: fixes f g :: "real => real" shows "[| 0 < d; ∀z. ¦z - x¦ ≤ d --> g(f(z)) = z; ∀z. ¦z - x¦ ≤ d --> isCont f z |] ==> ∃e. 0 < e & (∀y. 0 < ¦y - f(x)¦ & ¦y - f(x)¦ < e --> f(g(y)) = y)" apply (drule isCont_inj_range) prefer 2 apply (assumption, assumption, auto) apply (rule_tac x = e in exI, auto) apply (rotate_tac 2) apply (drule_tac x = y in spec, auto) done text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*} lemma LIM_fun_gt_zero: "[| f -- c --> (l::real); 0 < l |] ==> ∃r. 0 < r & (∀x::real. x ≠ c & ¦c - x¦ < r --> 0 < f x)" apply (auto simp add: LIM_def) apply (drule_tac x = "l/2" in spec, safe, force) apply (rule_tac x = s in exI) apply (auto simp only: abs_less_iff) done lemma LIM_fun_less_zero: "[| f -- c --> (l::real); l < 0 |] ==> ∃r. 0 < r & (∀x::real. x ≠ c & ¦c - x¦ < r --> f x < 0)" apply (auto simp add: LIM_def) apply (drule_tac x = "-l/2" in spec, safe, force) apply (rule_tac x = s in exI) apply (auto simp only: abs_less_iff) done lemma LIM_fun_not_zero: "[| f -- c --> (l::real); l ≠ 0 |] ==> ∃r. 0 < r & (∀x::real. x ≠ c & ¦c - x¦ < r --> f x ≠ 0)" apply (cut_tac x = l and y = 0 in linorder_less_linear, auto) apply (drule LIM_fun_less_zero) apply (drule_tac [3] LIM_fun_gt_zero) apply force+ done end
lemma lemma_realpow_diff:
p ≤ n ==> y ^ (Suc n - p) = y ^ (n - p) * y
lemma lemma_realpow_diff_sumr:
(∑p = 0..<Suc n. x ^ p * y ^ (Suc n - p)) =
y * (∑p = 0..<Suc n. x ^ p * y ^ (n - p))
lemma lemma_realpow_diff_sumr2:
x ^ Suc n - y ^ Suc n = (x - y) * (∑p = 0..<Suc n. x ^ p * y ^ (n - p))
lemma lemma_realpow_rev_sumr:
(∑p = 0..<Suc n. x ^ p * y ^ (n - p)) = (∑p = 0..<Suc n. x ^ (n - p) * y ^ p)
lemma powser_insidea:
[| summable (λn. f n * x ^ n); norm z < norm x |]
==> summable (λn. norm (f n * z ^ n))
lemma powser_inside:
[| summable (λn. f n * x ^ n); norm z < norm x |] ==> summable (λn. f n * z ^ n)
lemma diffs_minus:
diffs (λn. - c n) = (λn. - diffs c n)
lemma lemma_diffs:
(∑n = 0..<n. diffs c n * x ^ n) =
(∑n = 0..<n. of_nat n * c n * x ^ (n - Suc 0)) +
of_nat n * c n * x ^ (n - Suc 0)
lemma lemma_diffs2:
(∑n = 0..<n. of_nat n * c n * x ^ (n - Suc 0)) =
(∑n = 0..<n. diffs c n * x ^ n) - of_nat n * c n * x ^ (n - Suc 0)
lemma diffs_equiv:
summable (λn. diffs c n * x ^ n)
==> (λn. of_nat n * c n * x ^ (n - Suc 0)) sums (∑n. diffs c n * x ^ n)
lemma lemma_termdiff1:
(∑p = 0..<m. (z + h) ^ (m - p) * z ^ p - z ^ m) =
(∑p = 0..<m. z ^ p * ((z + h) ^ (m - p) - z ^ (m - p)))
lemma less_add_one:
m < n ==> ∃d. n = m + d + Suc 0
lemma sumdiff:
a + b - (c + d) = a - c + b - d
lemma sumr_diff_mult_const2:
setsum f {0..<n} - of_nat n * r = (∑i = 0..<n. f i - r)
lemma lemma_termdiff2:
h ≠ (0::'a)
==> ((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
h * (∑p = 0..<n - Suc 0.
∑q = 0..<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
lemma real_setsum_nat_ivl_bounded2:
[| !!p. p < n ==> f p ≤ K; (0::'a) ≤ K |]
==> setsum f {0..<n - k} ≤ of_nat n * K
lemma lemma_termdiff3:
[| h ≠ (0::'a); norm z ≤ K; norm (z + h) ≤ K |]
==> norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
≤ real_of_nat n * real_of_nat (n - Suc 0) * K ^ (n - 2) * norm h
lemma lemma_termdiff4:
[| 0 < k; !!h. [| h ≠ (0::'a); norm h < k |] ==> norm (f h) ≤ K * norm h |]
==> f -- 0::'a --> (0::'b)
lemma lemma_termdiff5:
[| 0 < k; summable f;
!!h n. [| h ≠ (0::'a); norm h < k |] ==> norm (g h n) ≤ f n * norm h |]
==> (λh. suminf (g h)) -- 0::'a --> (0::'b)
lemma termdiffs_aux:
[| summable (λn. diffs (diffs c) n * K ^ n); norm x < norm K |]
==> (λh. ∑n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
-- 0::'a --> (0::'a)
lemma termdiffs:
[| summable (λn. c n * K ^ n); summable (λn. diffs c n * K ^ n);
summable (λn. diffs (diffs c) n * K ^ n); norm x < norm K |]
==> DERIV (λx. ∑n. c n * x ^ n) x :> (∑n. diffs c n * x ^ n)
lemma summable_exp_generic:
summable (λn. x ^ n /R real (fact n))
lemma summable_norm_exp:
summable (λn. norm (x ^ n /R real (fact n)))
lemma summable_exp:
summable (λn. inverse (real (fact n)) * x ^ n)
lemma summable_sin:
summable
(λn. (if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n)) * x ^ n)
lemma summable_cos:
summable (λn. (if even n then -1 ^ (n div 2) / real (fact n) else 0) * x ^ n)
lemma lemma_STAR_sin:
(if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n)) * 0 ^ n = 0
lemma lemma_STAR_cos:
0 < n --> -1 ^ (n div 2) / real (fact n) * 0 ^ n = 0
lemma lemma_STAR_cos1:
0 < n --> -1 ^ (n div 2) / real (fact n) * 0 ^ n = 0
lemma lemma_STAR_cos2:
(∑n = 1..<n. if even n then -1 ^ (n div 2) / real (fact n) * 0 ^ n else 0) = 0
lemma exp_converges:
(λn. x ^ n /R real (fact n)) sums exp x
lemma sin_converges:
(λn. (if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n)) *
x ^ n) sums
sin x
lemma cos_converges:
(λn. (if even n then -1 ^ (n div 2) / real (fact n) else 0) * x ^ n) sums cos x
lemma exp_fdiffs:
diffs (λn. inverse (real (fact n))) = (λn. inverse (real (fact n)))
lemma diffs_of_real:
diffs (λn. of_real (f n)) = (λn. of_real (diffs f n))
lemma sin_fdiffs:
diffs (λn. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n)) =
(λn. if even n then -1 ^ (n div 2) / real (fact n) else 0)
lemma sin_fdiffs2:
diffs (λn. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n)) n =
(if even n then -1 ^ (n div 2) / real (fact n) else 0)
lemma cos_fdiffs:
diffs (λn. if even n then -1 ^ (n div 2) / real (fact n) else 0) =
(λn. - (if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n)))
lemma cos_fdiffs2:
diffs (λn. if even n then -1 ^ (n div 2) / real (fact n) else 0) n =
- (if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))
lemma lemma_sin_minus:
- sin x =
(∑n. - ((if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n)) *
x ^ n))
lemma lemma_exp_ext:
exp = (λx. ∑n. x ^ n /R real (fact n))
lemma DERIV_exp:
DERIV exp x :> exp x
lemma lemma_sin_ext:
sin =
(λx. ∑n. (if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n)) *
x ^ n)
lemma lemma_cos_ext:
cos = (λx. ∑n. (if even n then -1 ^ (n div 2) / real (fact n) else 0) * x ^ n)
lemma DERIV_sin:
DERIV sin x :> cos x
lemma DERIV_cos:
DERIV cos x :> - sin x
lemma isCont_exp:
isCont exp x
lemma isCont_sin:
isCont sin x
lemma isCont_cos:
isCont cos x
lemma powser_zero:
(∑n. f n * (0::'a) ^ n) = f 0
lemma exp_zero:
exp (0::'a) = (1::'a)
lemma setsum_head2:
m ≤ n ==> setsum f {m..n} = f m + setsum f {Suc m..n}
lemma setsum_cl_ivl_Suc2:
setsum f {m..Suc n} =
(if Suc n < m then 0::'a else f m + (∑i = m..n. f (Suc i)))
lemma exp_series_add:
(x + y) ^ n /R real (fact n) =
(∑i = 0..n. x ^ i /R real (fact i) * (y ^ (n - i) /R real (fact (n - i))))
lemma exp_add:
exp (x + y) = exp x * exp y
lemma exp_of_real:
exp (of_real x) = of_real (exp x)
lemma exp_ge_add_one_self_aux:
0 ≤ x ==> 1 + x ≤ exp x
lemma exp_gt_one:
0 < x ==> 1 < exp x
lemma DERIV_exp_add_const:
DERIV (λx. exp (x + y)) x :> exp (x + y)
lemma DERIV_exp_minus:
DERIV (λx. exp (- x)) x :> - exp (- x)
lemma DERIV_exp_exp_zero:
DERIV (λx. exp (x + y) * exp (- x)) x :> (0::'a)
lemma exp_add_mult_minus:
exp (x + y) * exp (- x) = exp y
lemma exp_mult_minus:
exp x * exp (- x) = (1::'a)
lemma exp_mult_minus2:
exp (- x) * exp x = (1::'a)
lemma exp_minus:
exp (- x) = inverse (exp x)
lemma exp_ge_zero:
0 ≤ exp x
lemma exp_not_eq_zero:
exp x ≠ (0::'a)
lemma exp_gt_zero:
0 < exp x
lemma inv_exp_gt_zero:
0 < inverse (exp x)
lemma abs_exp_cancel:
¦exp x¦ = exp x
lemma exp_real_of_nat_mult:
exp (real n * x) = exp x ^ n
lemma exp_diff:
exp (x - y) = exp x / exp y
lemma exp_less_mono:
x < y ==> exp x < exp y
lemma exp_less_cancel:
exp x < exp y ==> x < y
lemma exp_less_cancel_iff:
(exp x < exp y) = (x < y)
lemma exp_le_cancel_iff:
(exp x ≤ exp y) = (x ≤ y)
lemma exp_inj_iff:
(exp x = exp y) = (x = y)
lemma lemma_exp_total:
1 ≤ y ==> ∃x≥0. x ≤ y - 1 ∧ exp x = y
lemma exp_total:
0 < y ==> ∃x. exp x = y
lemma ln_exp:
ln (exp x) = x
lemma exp_ln:
0 < x ==> exp (ln x) = x
lemma exp_ln_iff:
(exp (ln x) = x) = (0 < x)
lemma ln_mult:
[| 0 < x; 0 < y |] ==> ln (x * y) = ln x + ln y
lemma ln_inj_iff:
[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)
lemma ln_one:
ln 1 = 0
lemma ln_inverse:
0 < x ==> ln (inverse x) = - ln x
lemma ln_div:
[| 0 < x; 0 < y |] ==> ln (x / y) = ln x - ln y
lemma ln_less_cancel_iff:
[| 0 < x; 0 < y |] ==> (ln x < ln y) = (x < y)
lemma ln_le_cancel_iff:
[| 0 < x; 0 < y |] ==> (ln x ≤ ln y) = (x ≤ y)
lemma ln_realpow:
0 < x ==> ln (x ^ n) = real n * ln x
lemma ln_add_one_self_le_self:
0 ≤ x ==> ln (1 + x) ≤ x
lemma ln_less_self:
0 < x ==> ln x < x
lemma ln_ge_zero:
1 ≤ x ==> 0 ≤ ln x
lemma ln_ge_zero_imp_ge_one:
[| 0 ≤ ln x; 0 < x |] ==> 1 ≤ x
lemma ln_ge_zero_iff:
0 < x ==> (0 ≤ ln x) = (1 ≤ x)
lemma ln_less_zero_iff:
0 < x ==> (ln x < 0) = (x < 1)
lemma ln_gt_zero:
1 < x ==> 0 < ln x
lemma ln_gt_zero_imp_gt_one:
[| 0 < ln x; 0 < x |] ==> 1 < x
lemma ln_gt_zero_iff:
0 < x ==> (0 < ln x) = (1 < x)
lemma ln_eq_zero_iff:
0 < x ==> (ln x = 0) = (x = 1)
lemma ln_less_zero:
[| 0 < x; x < 1 |] ==> ln x < 0
lemma exp_ln_eq:
exp u = x ==> ln x = u
lemma isCont_ln:
0 < x ==> isCont ln x
lemma DERIV_ln:
0 < x ==> DERIV ln x :> inverse x
lemma sin_zero:
sin 0 = 0
lemma cos_zero:
cos 0 = 1
lemma DERIV_sin_sin_mult:
DERIV (λx. sin x * sin x) x :> cos x * sin x + cos x * sin x
lemma DERIV_sin_sin_mult2:
DERIV (λx. sin x * sin x) x :> 2 * cos x * sin x
lemma DERIV_sin_realpow2:
DERIV (λx. sin x ^ 2) x :> cos x * sin x + cos x * sin x
lemma DERIV_sin_realpow2a:
DERIV (λx. sin x ^ 2) x :> 2 * cos x * sin x
lemma DERIV_cos_cos_mult:
DERIV (λx. cos x * cos x) x :> - sin x * cos x + - sin x * cos x
lemma DERIV_cos_cos_mult2:
DERIV (λx. cos x * cos x) x :> -2 * cos x * sin x
lemma DERIV_cos_realpow2:
DERIV (λx. cos x ^ 2) x :> - sin x * cos x + - sin x * cos x
lemma DERIV_cos_realpow2a:
DERIV (λx. cos x ^ 2) x :> -2 * cos x * sin x
lemma lemma_DERIV_subst:
[| DERIV f x :> D; D = E |] ==> DERIV f x :> E
lemma DERIV_cos_realpow2b:
DERIV (λx. cos x ^ 2) x :> - (2 * cos x * sin x)
lemma DERIV_cos_cos_mult3:
DERIV (λx. cos x * cos x) x :> - (2 * cos x * sin x)
lemma DERIV_sin_circle_all:
∀x. DERIV (λx. sin x ^ 2 + cos x ^ 2) x :> 2 * cos x * sin x - 2 * cos x * sin x
lemma DERIV_sin_circle_all_zero:
∀x. DERIV (λx. sin x ^ 2 + cos x ^ 2) x :> 0
lemma sin_cos_squared_add:
sin x ^ 2 + cos x ^ 2 = 1
lemma sin_cos_squared_add2:
cos x ^ 2 + sin x ^ 2 = 1
lemma sin_cos_squared_add3:
cos x * cos x + sin x * sin x = 1
lemma sin_squared_eq:
sin x ^ 2 = 1 - cos x ^ 2
lemma cos_squared_eq:
cos x ^ 2 = 1 - sin x ^ 2
lemma real_gt_one_ge_zero_add_less:
[| 1 < x; 0 ≤ y |] ==> 1 < x + y
lemma abs_sin_le_one:
¦sin x¦ ≤ 1
lemma sin_ge_minus_one:
-1 ≤ sin x
lemma sin_le_one:
sin x ≤ 1
lemma abs_cos_le_one:
¦cos x¦ ≤ 1
lemma cos_ge_minus_one:
-1 ≤ cos x
lemma cos_le_one:
cos x ≤ 1
lemma DERIV_fun_pow:
DERIV g x :> m ==> DERIV (λx. g x ^ n) x :> real n * g x ^ (n - 1) * m
lemma DERIV_fun_exp:
DERIV g x :> m ==> DERIV (λx. exp (g x)) x :> exp (g x) * m
lemma DERIV_fun_sin:
DERIV g x :> m ==> DERIV (λx. sin (g x)) x :> cos (g x) * m
lemma DERIV_fun_cos:
DERIV g x :> m ==> DERIV (λx. cos (g x)) x :> - sin (g x) * m
lemma DERIV_intros:
DERIV (λx. x) x :> (1::'a)
DERIV (λx. k) x :> (0::'a)
DERIV cos x :> - sin x
DERIV f x :> D ==> DERIV (λx. c * f x) x :> c * D
DERIV sin x :> cos x
DERIV exp x :> exp x
x ≠ (0::'a) ==> DERIV inverse x :> - (inverse x ^ Suc (Suc 0))
DERIV (λx. x ^ n) x :> real n * x ^ (n - Suc 0)
[| DERIV f x :> D; DERIV g x :> E |] ==> DERIV (λx. f x + g x) x :> D + E
[| DERIV f x :> D; DERIV g x :> E |] ==> DERIV (λx. f x - g x) x :> D - E
[| DERIV f x :> Da; DERIV g x :> Db |]
==> DERIV (λx. f x * g x) x :> Da * g x + Db * f x
DERIV f x :> D ==> DERIV (λx. - f x) x :> - D
[| DERIV f x :> d; f x ≠ (0::'a) |]
==> DERIV (λx. inverse (f x)) x :> - (d * inverse (f x ^ Suc (Suc 0)))
[| DERIV f x :> d; DERIV g x :> e; g x ≠ (0::'a) |]
==> DERIV (λy. f y / g y) x :> (d * g x - e * f x) / g x ^ Suc (Suc 0)
DERIV g x :> m ==> DERIV (λx. g x ^ n) x :> real n * g x ^ (n - 1) * m
DERIV g x :> m ==> DERIV (λx. exp (g x)) x :> exp (g x) * m
DERIV g x :> m ==> DERIV (λx. sin (g x)) x :> cos (g x) * m
DERIV g x :> m ==> DERIV (λx. cos (g x)) x :> - sin (g x) * m
lemma lemma_DERIV_sin_cos_add:
∀x. DERIV (λx. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2)
x :> 0
lemma sin_cos_add:
(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 =
0
lemma sin_add:
sin (x + y) = sin x * cos y + cos x * sin y
lemma cos_add:
cos (x + y) = cos x * cos y - sin x * sin y
lemma lemma_DERIV_sin_cos_minus:
∀x. DERIV (λx. (sin (- x) + sin x) ^ 2 + (cos (- x) - cos x) ^ 2) x :> 0
lemma sin_cos_minus:
(sin (- x) + sin x) ^ 2 + (cos (- x) - cos x) ^ 2 = 0
lemma sin_minus:
sin (- x) = - sin x
lemma cos_minus:
cos (- x) = cos x
lemma sin_diff:
sin (x - y) = sin x * cos y - cos x * sin y
lemma sin_diff2:
sin (x - y) = cos y * sin x - sin y * cos x
lemma cos_diff:
cos (x - y) = cos x * cos y + sin x * sin y
lemma cos_diff2:
cos (x - y) = cos y * cos x + sin y * sin x
lemma sin_double:
sin (2 * x) = 2 * sin x * cos x
lemma cos_double:
cos (2 * x) = cos x ^ 2 - sin x ^ 2
lemma sin_paired:
(λn. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums sin x
lemma sin_gt_zero:
[| 0 < x; x < 2 |] ==> 0 < sin x
lemma sin_gt_zero1:
[| 0 < x; x < 2 |] ==> 0 < sin x
lemma cos_double_less_one:
[| 0 < x; x < 2 |] ==> cos (2 * x) < 1
lemma cos_paired:
(λn. -1 ^ n / real (fact (2 * n)) * x ^ (2 * n)) sums cos x
lemma fact_lemma:
real n * 4 = real (4 * n)
lemma cos_two_less_zero:
cos 2 < 0
lemma cos_two_neq_zero:
cos 2 ≠ 0
lemma cos_two_le_zero:
cos 2 ≤ 0
lemma cos_is_zero:
∃!x. 0 ≤ x ∧ x ≤ 2 ∧ cos x = 0
lemma pi_half:
pi / 2 = (THE x. 0 ≤ x ∧ x ≤ 2 ∧ cos x = 0)
lemma cos_pi_half:
cos (pi / 2) = 0
lemma pi_half_gt_zero:
0 < pi / 2
lemma pi_half_neq_zero:
pi / 2 ≠ 0
lemma pi_half_ge_zero:
0 ≤ pi / 2
lemma pi_half_less_two:
pi / 2 < 2
lemma pi_half_neq_two:
pi / 2 ≠ 2
lemma pi_half_le_two:
pi / 2 ≤ 2
lemma pi_gt_zero:
0 < pi
lemma pi_ge_zero:
0 ≤ pi
lemma pi_neq_zero:
pi ≠ 0
lemma pi_not_less_zero:
¬ pi < 0
lemma minus_pi_half_less_zero:
- (pi / 2) < 0
lemma sin_pi_half:
sin (pi / 2) = 1
lemma cos_pi:
cos pi = -1
lemma sin_pi:
sin pi = 0
lemma sin_cos_eq:
sin x = cos (pi / 2 - x)
lemma minus_sin_cos_eq:
- sin x = cos (x + pi / 2)
lemma cos_sin_eq:
cos x = sin (pi / 2 - x)
lemma sin_periodic_pi:
sin (x + pi) = - sin x
lemma sin_periodic_pi2:
sin (pi + x) = - sin x
lemma cos_periodic_pi:
cos (x + pi) = - cos x
lemma sin_periodic:
sin (x + 2 * pi) = sin x
lemma cos_periodic:
cos (x + 2 * pi) = cos x
lemma cos_npi:
cos (real n * pi) = -1 ^ n
lemma cos_npi2:
cos (pi * real n) = -1 ^ n
lemma sin_npi:
sin (real n * pi) = 0
lemma sin_npi2:
sin (pi * real n) = 0
lemma cos_two_pi:
cos (2 * pi) = 1
lemma sin_two_pi:
sin (2 * pi) = 0
lemma sin_gt_zero2:
[| 0 < x; x < pi / 2 |] ==> 0 < sin x
lemma sin_less_zero:
[| - pi / 2 < x; x < 0 |] ==> sin x < 0
lemma pi_less_4:
pi < 4
lemma cos_gt_zero:
[| 0 < x; x < pi / 2 |] ==> 0 < cos x
lemma cos_gt_zero_pi:
[| - (pi / 2) < x; x < pi / 2 |] ==> 0 < cos x
lemma cos_ge_zero:
[| - (pi / 2) ≤ x; x ≤ pi / 2 |] ==> 0 ≤ cos x
lemma sin_gt_zero_pi:
[| 0 < x; x < pi |] ==> 0 < sin x
lemma sin_ge_zero:
[| 0 ≤ x; x ≤ pi |] ==> 0 ≤ sin x
lemma cos_total:
[| -1 ≤ y; y ≤ 1 |] ==> ∃!x. 0 ≤ x ∧ x ≤ pi ∧ cos x = y
lemma sin_total:
[| -1 ≤ y; y ≤ 1 |] ==> ∃!x. - (pi / 2) ≤ x ∧ x ≤ pi / 2 ∧ sin x = y
lemma reals_Archimedean4:
[| 0 < y; 0 ≤ x |] ==> ∃n. real n * y ≤ x ∧ x < real (Suc n) * y
lemma cos_zero_lemma:
[| 0 ≤ x; cos x = 0 |] ==> ∃n. odd n ∧ x = real n * (pi / 2)
lemma sin_zero_lemma:
[| 0 ≤ x; sin x = 0 |] ==> ∃n. even n ∧ x = real n * (pi / 2)
lemma cos_zero_iff:
(cos x = 0) =
((∃n. odd n ∧ x = real n * (pi / 2)) ∨ (∃n. odd n ∧ x = - (real n * (pi / 2))))
lemma sin_zero_iff:
(sin x = 0) =
((∃n. even n ∧ x = real n * (pi / 2)) ∨
(∃n. even n ∧ x = - (real n * (pi / 2))))
lemma tan_zero:
tan 0 = 0
lemma tan_pi:
tan pi = 0
lemma tan_npi:
tan (real n * pi) = 0
lemma tan_minus:
tan (- x) = - tan x
lemma tan_periodic:
tan (x + 2 * pi) = tan x
lemma lemma_tan_add1:
[| cos x ≠ 0; cos y ≠ 0 |] ==> 1 - tan x * tan y = cos (x + y) / (cos x * cos y)
lemma add_tan_eq:
[| cos x ≠ 0; cos y ≠ 0 |] ==> tan x + tan y = sin (x + y) / (cos x * cos y)
lemma tan_add:
[| cos x ≠ 0; cos y ≠ 0; cos (x + y) ≠ 0 |]
==> tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)
lemma tan_double:
[| cos x ≠ 0; cos (2 * x) ≠ 0 |] ==> tan (2 * x) = 2 * tan x / (1 - tan x ^ 2)
lemma tan_gt_zero:
[| 0 < x; x < pi / 2 |] ==> 0 < tan x
lemma tan_less_zero:
[| - pi / 2 < x; x < 0 |] ==> tan x < 0
lemma lemma_DERIV_tan:
cos x ≠ 0 ==> DERIV (λx. sin x / cos x) x :> inverse (cos x ^ 2)
lemma DERIV_tan:
cos x ≠ 0 ==> DERIV tan x :> inverse (cos x ^ 2)
lemma isCont_tan:
cos x ≠ 0 ==> isCont tan x
lemma LIM_cos_div_sin:
(λx. cos x / sin x) -- pi / 2 --> 0
lemma lemma_tan_total:
0 < y ==> ∃x>0. x < pi / 2 ∧ y < tan x
lemma tan_total_pos:
0 ≤ y ==> ∃x≥0. x < pi / 2 ∧ tan x = y
lemma lemma_tan_total1:
∃x>- (pi / 2). x < pi / 2 ∧ tan x = y
lemma tan_total:
∃!x. - (pi / 2) < x ∧ x < pi / 2 ∧ tan x = y
lemma arcsin:
[| -1 ≤ y; y ≤ 1 |]
==> - (pi / 2) ≤ arcsin y ∧ arcsin y ≤ pi / 2 ∧ sin (arcsin y) = y
lemma arcsin_pi:
[| -1 ≤ y; y ≤ 1 |]
==> - (pi / 2) ≤ arcsin y ∧ arcsin y ≤ pi ∧ sin (arcsin y) = y
lemma sin_arcsin:
[| -1 ≤ y; y ≤ 1 |] ==> sin (arcsin y) = y
lemma arcsin_bounded:
[| -1 ≤ y; y ≤ 1 |] ==> - (pi / 2) ≤ arcsin y ∧ arcsin y ≤ pi / 2
lemma arcsin_lbound:
[| -1 ≤ y; y ≤ 1 |] ==> - (pi / 2) ≤ arcsin y
lemma arcsin_ubound:
[| -1 ≤ y; y ≤ 1 |] ==> arcsin y ≤ pi / 2
lemma arcsin_lt_bounded:
[| -1 < y; y < 1 |] ==> - (pi / 2) < arcsin y ∧ arcsin y < pi / 2
lemma arcsin_sin:
[| - (pi / 2) ≤ x; x ≤ pi / 2 |] ==> arcsin (sin x) = x
lemma arccos:
[| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arccos y ∧ arccos y ≤ pi ∧ cos (arccos y) = y
lemma cos_arccos:
[| -1 ≤ y; y ≤ 1 |] ==> cos (arccos y) = y
lemma arccos_bounded:
[| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arccos y ∧ arccos y ≤ pi
lemma arccos_lbound:
[| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arccos y
lemma arccos_ubound:
[| -1 ≤ y; y ≤ 1 |] ==> arccos y ≤ pi
lemma arccos_lt_bounded:
[| -1 < y; y < 1 |] ==> 0 < arccos y ∧ arccos y < pi
lemma arccos_cos:
[| 0 ≤ x; x ≤ pi |] ==> arccos (cos x) = x
lemma arccos_cos2:
[| x ≤ 0; - pi ≤ x |] ==> arccos (cos x) = - x
lemma cos_arcsin:
[| -1 ≤ x; x ≤ 1 |] ==> cos (arcsin x) = sqrt (1 - x ^ 2)
lemma sin_arccos:
[| -1 ≤ x; x ≤ 1 |] ==> sin (arccos x) = sqrt (1 - x ^ 2)
lemma arctan:
- (pi / 2) < arctan y ∧ arctan y < pi / 2 ∧ tan (arctan y) = y
lemma tan_arctan:
tan (arctan y) = y
lemma arctan_bounded:
- (pi / 2) < arctan y ∧ arctan y < pi / 2
lemma arctan_lbound:
- (pi / 2) < arctan y
lemma arctan_ubound:
arctan y < pi / 2
lemma arctan_tan:
[| - (pi / 2) < x; x < pi / 2 |] ==> arctan (tan x) = x
lemma arctan_zero_zero:
arctan 0 = 0
lemma cos_arctan_not_zero:
cos (arctan x) ≠ 0
lemma tan_sec:
cos x ≠ 0 ==> 1 + tan x ^ 2 = inverse (cos x) ^ 2
lemma isCont_inverse_function2:
[| a < x; x < b; ∀z. a ≤ z ∧ z ≤ b --> g (f z) = z;
∀z. a ≤ z ∧ z ≤ b --> isCont f z |]
==> isCont g (f x)
lemma isCont_arcsin:
[| -1 < x; x < 1 |] ==> isCont arcsin x
lemma isCont_arccos:
[| -1 < x; x < 1 |] ==> isCont arccos x
lemma isCont_arctan:
isCont arctan x
lemma DERIV_arcsin:
[| -1 < x; x < 1 |] ==> DERIV arcsin x :> inverse (sqrt (1 - x ^ 2))
lemma DERIV_arccos:
[| -1 < x; x < 1 |] ==> DERIV arccos x :> inverse (- sqrt (1 - x ^ 2))
lemma DERIV_arctan:
DERIV arctan x :> inverse (1 + x ^ 2)
lemma cos_45:
cos (pi / 4) = sqrt 2 / 2
lemma cos_30:
cos (pi / 6) = sqrt 3 / 2
lemma sin_45:
sin (pi / 4) = sqrt 2 / 2
lemma sin_60:
sin (pi / 3) = sqrt 3 / 2
lemma cos_60:
cos (pi / 3) = 1 / 2
lemma sin_30:
sin (pi / 6) = 1 / 2
lemma tan_30:
tan (pi / 6) = 1 / sqrt 3
lemma tan_45:
tan (pi / 4) = 1
lemma tan_60:
tan (pi / 3) = sqrt 3
lemma
sin (x + 1 / 2 * real (Suc m) * pi) = cos (x + 1 / 2 * real m * pi)
lemma
sin (x + real (Suc m) * pi / 2) = cos (x + real m * pi / 2)
lemma DERIV_sin_add:
DERIV (λx. sin (x + k)) xa :> cos (xa + k)
lemma sin_cos_npi:
sin (real (Suc (2 * n)) * pi / 2) = -1 ^ n
lemma cos_2npi:
cos (2 * real n * pi) = 1
lemma cos_3over2_pi:
cos (3 / 2 * pi) = 0
lemma sin_2npi:
sin (2 * real n * pi) = 0
lemma sin_3over2_pi:
sin (3 / 2 * pi) = - 1
lemma
cos (x + 1 / 2 * real (Suc m) * pi) = - sin (x + 1 / 2 * real m * pi)
lemma
cos (x + real (Suc m) * pi / 2) = - sin (x + real m * pi / 2)
lemma cos_pi_eq_zero:
cos (pi * real (Suc (2 * m)) / 2) = 0
lemma DERIV_cos_add:
DERIV (λx. cos (x + k)) xa :> - sin (xa + k)
lemma sin_zero_abs_cos_one:
sin x = 0 ==> ¦cos x¦ = 1
lemma exp_eq_one_iff:
(exp x = 1) = (x = 0)
lemma cos_one_sin_zero:
cos x = 1 ==> sin x = 0
lemma cos_x_y_le_one:
¦x / sqrt (x ^ 2 + y ^ 2)¦ ≤ 1
lemma cos_arccos_abs:
¦y¦ ≤ 1 ==> cos (arccos y) = y
lemma sin_arccos_abs:
¦y¦ ≤ 1 ==> sin (arccos y) = sqrt (1 - y ^ 2)
lemma cos_arccos_lemma1:
cos (arccos (x1 / sqrt (x1 ^ 2 + y1 ^ 2))) = x1 / sqrt (x1 ^ 2 + y1 ^ 2)
lemma sin_arccos_lemma1:
sin (arccos (x1 / sqrt (x1 ^ 2 + y1 ^ 2))) =
sqrt (1 - (x1 / sqrt (x1 ^ 2 + y1 ^ 2)) ^ 2)
lemma polar_ex1:
0 < y ==> ∃r a. x = r * cos a ∧ y = r * sin a
lemma polar_ex2:
y < 0 ==> ∃r a. x = r * cos a ∧ y = r * sin a
lemma polar_Ex:
∃r a. x = r * cos a ∧ y = r * sin a
lemma isCont_inv_fun:
[| 0 < d; ∀z. ¦z - x¦ ≤ d --> g (f z) = z; ∀z. ¦z - x¦ ≤ d --> isCont f z |]
==> isCont g (f x)
lemma isCont_inv_fun_inv:
[| 0 < d; ∀z. ¦z - x¦ ≤ d --> g (f z) = z; ∀z. ¦z - x¦ ≤ d --> isCont f z |]
==> ∃e>0. ∀y. 0 < ¦y - f x¦ ∧ ¦y - f x¦ < e --> f (g y) = y
lemma LIM_fun_gt_zero:
[| f -- c --> l; 0 < l |] ==> ∃r>0. ∀x. x ≠ c ∧ ¦c - x¦ < r --> 0 < f x
lemma LIM_fun_less_zero:
[| f -- c --> l; l < 0 |] ==> ∃r>0. ∀x. x ≠ c ∧ ¦c - x¦ < r --> f x < 0
lemma LIM_fun_not_zero:
[| f -- c --> l; l ≠ 0 |] ==> ∃r>0. ∀x. x ≠ c ∧ ¦c - x¦ < r --> f x ≠ 0