# Theory Transcendental

Up to index of Isabelle/HOL

theory Transcendental
imports Series NthRoot
`(*  Title:      HOL/Transcendental.thy    Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh    Author:     Lawrence C Paulson*)header{*Power Series, Transcendental Functions etc.*}theory Transcendentalimports Fact Series Deriv NthRootbeginsubsection {* Properties of Power Series *}lemma lemma_realpow_diff:  fixes y :: "'a::monoid_mult"  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_commutes)qedlemma lemma_realpow_diff_sumr:  fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows     "(∑p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =      y * (∑p=0..<Suc n. (x ^ p) * y ^ (n - p))"by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac         del: setsum_op_ivl_Suc)lemma lemma_realpow_diff_sumr2:  fixes y :: "'a::{comm_ring,monoid_mult}" shows     "x ^ (Suc n) - y ^ (Suc n) =      (x - y) * (∑p=0..<Suc n. (x ^ p) * y ^ (n - p))"apply (induct n, simp)apply (simp del: setsum_op_ivl_Suc)apply (subst setsum_op_ivl_Suc)apply (subst lemma_realpow_diff_sumr)apply (simp add: distrib_left del: setsum_op_ivl_Suc)apply (subst mult_left_commute [of "x - y"])apply (erule subst)apply (simp add: algebra_simps)donelemma 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 autoapply (rule_tac x="n - x" in image_eqI, simp, simp)donetext{*Power series has a `circle` of convergence, i.e. if it sums for @{termx}, then it sums absolutely for @{term z} with @{term "¦z¦ < ¦x¦"}.*}lemma powser_insidea:  fixes x z :: "'a::real_normed_field"  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 (rule convergent_Cauchy)  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)qedlemma powser_inside:  fixes f :: "nat => 'a::{real_normed_field,banach}" 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])lemma sum_split_even_odd: fixes f :: "nat => real" shows  "(∑ i = 0 ..< 2 * n. if even i then f i else g i) =   (∑ i = 0 ..< n. f (2 * i)) + (∑ i = 0 ..< n. g (2 * i + 1))"proof (induct n)  case (Suc n)  have "(∑ i = 0 ..< 2 * Suc n. if even i then f i else g i) =        (∑ i = 0 ..< n. f (2 * i)) + (∑ i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"    using Suc.hyps unfolding One_nat_def by auto  also have "… = (∑ i = 0 ..< Suc n. f (2 * i)) + (∑ i = 0 ..< Suc n. g (2 * i + 1))" by auto  finally show ?case .qed autolemma sums_if': fixes g :: "nat => real" assumes "g sums x"  shows "(λ n. if even n then 0 else g ((n - 1) div 2)) sums x"  unfolding sums_defproof (rule LIMSEQ_I)  fix r :: real assume "0 < r"  from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]  obtain no where no_eq: "!! n. n ≥ no ==> (norm (setsum g { 0..<n } - x) < r)" by blast  let ?SUM = "λ m. ∑ i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"  { fix m assume "m ≥ 2 * no" hence "m div 2 ≥ no" by auto    have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"      using sum_split_even_odd by auto    hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 ≥ no` by auto    moreover    have "?SUM (2 * (m div 2)) = ?SUM m"    proof (cases "even m")      case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..    next      case False hence "even (Suc m)" by auto      from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]      have eq: "Suc (2 * (m div 2)) = m" by auto      hence "even (2 * (m div 2))" using `odd m` by auto      have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..      also have "… = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto      finally show ?thesis by auto    qed    ultimately have "(norm (?SUM m - x) < r)" by auto  }  thus "∃ no. ∀ m ≥ no. norm (?SUM m - x) < r" by blastqedlemma sums_if: fixes g :: "nat => real" assumes "g sums x" and "f sums y"  shows "(λ n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"proof -  let ?s = "λ n. if even n then 0 else f ((n - 1) div 2)"  { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"      by (cases B) auto } note if_sum = this  have g_sums: "(λ n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .  {    have "?s 0 = 0" by auto    have Suc_m1: "!! n. Suc n - 1 = n" by auto    have if_eq: "!!B T E. (if ¬ B then T else E) = (if B then E else T)" by auto    have "?s sums y" using sums_if'[OF `f sums y`] .    from this[unfolded sums_def, THEN LIMSEQ_Suc]    have "(λ n. if even n then f (n div 2) else 0) sums y"      unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]                image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]                even_Suc Suc_m1 if_eq .  } from sums_add[OF g_sums this]  show ?thesis unfolding if_sum .qedsubsection {* Alternating series test / Leibniz formula *}lemma sums_alternating_upper_lower:  fixes a :: "nat => real"  assumes mono: "!!n. a (Suc n) ≤ a n" and a_pos: "!!n. 0 ≤ a n" and "a ----> 0"  shows "∃l. ((∀n. (∑i=0..<2*n. -1^i*a i) ≤ l) ∧ (λ n. ∑i=0..<2*n. -1^i*a i) ----> l) ∧             ((∀n. l ≤ (∑i=0..<2*n + 1. -1^i*a i)) ∧ (λ n. ∑i=0..<2*n + 1. -1^i*a i) ----> l)"  (is "∃l. ((∀n. ?f n ≤ l) ∧ _) ∧ ((∀n. l ≤ ?g n) ∧ _)")proof -  have fg_diff: "!!n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto  have "∀ n. ?f n ≤ ?f (Suc n)"  proof fix n show "?f n ≤ ?f (Suc n)" using mono[of "2*n"] by auto qed  moreover  have "∀ n. ?g (Suc n) ≤ ?g n"  proof fix n show "?g (Suc n) ≤ ?g n" using mono[of "Suc (2*n)"]    unfolding One_nat_def by auto qed  moreover  have "∀ n. ?f n ≤ ?g n"  proof fix n show "?f n ≤ ?g n" using fg_diff a_pos    unfolding One_nat_def by auto qed  moreover  have "(λ n. ?f n - ?g n) ----> 0" unfolding fg_diff  proof (rule LIMSEQ_I)    fix r :: real assume "0 < r"    with `a ----> 0`[THEN LIMSEQ_D]    obtain N where "!! n. n ≥ N ==> norm (a n - 0) < r" by auto    hence "∀ n ≥ N. norm (- a (2 * n) - 0) < r" by auto    thus "∃ N. ∀ n ≥ N. norm (- a (2 * n) - 0) < r" by auto  qed  ultimately  show ?thesis by (rule lemma_nest_unique)qedlemma summable_Leibniz': fixes a :: "nat => real"  assumes a_zero: "a ----> 0" and a_pos: "!! n. 0 ≤ a n"  and a_monotone: "!! n. a (Suc n) ≤ a n"  shows summable: "summable (λ n. (-1)^n * a n)"  and "!!n. (∑i=0..<2*n. (-1)^i*a i) ≤ (∑i. (-1)^i*a i)"  and "(λn. ∑i=0..<2*n. (-1)^i*a i) ----> (∑i. (-1)^i*a i)"  and "!!n. (∑i. (-1)^i*a i) ≤ (∑i=0..<2*n+1. (-1)^i*a i)"  and "(λn. ∑i=0..<2*n+1. (-1)^i*a i) ----> (∑i. (-1)^i*a i)"proof -  let "?S n" = "(-1)^n * a n"  let "?P n" = "∑i=0..<n. ?S i"  let "?f n" = "?P (2 * n)"  let "?g n" = "?P (2 * n + 1)"  obtain l :: real where below_l: "∀ n. ?f n ≤ l" and "?f ----> l" and above_l: "∀ n. l ≤ ?g n" and "?g ----> l"    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast  let ?Sa = "λ m. ∑ n = 0..<m. ?S n"  have "?Sa ----> l"  proof (rule LIMSEQ_I)    fix r :: real assume "0 < r"    with `?f ----> l`[THEN LIMSEQ_D]    obtain f_no where f: "!! n. n ≥ f_no ==> norm (?f n - l) < r" by auto    from `0 < r` `?g ----> l`[THEN LIMSEQ_D]    obtain g_no where g: "!! n. n ≥ g_no ==> norm (?g n - l) < r" by auto    { fix n :: nat      assume "n ≥ (max (2 * f_no) (2 * g_no))" hence "n ≥ 2 * f_no" and "n ≥ 2 * g_no" by auto      have "norm (?Sa n - l) < r"      proof (cases "even n")        case True from even_nat_div_two_times_two[OF this]        have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto        with `n ≥ 2 * f_no` have "n div 2 ≥ f_no" by auto        from f[OF this]        show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .      next        case False hence "even (n - 1)" by simp        from even_nat_div_two_times_two[OF this]        have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto        hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto        from n_eq `n ≥ 2 * g_no` have "(n - 1) div 2 ≥ g_no" by auto        from g[OF this]        show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .      qed    }    thus "∃ no. ∀ n ≥ no. norm (?Sa n - l) < r" by blast  qed  hence sums_l: "(λi. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .  thus "summable ?S" using summable_def by auto  have "l = suminf ?S" using sums_unique[OF sums_l] .  { fix n show "suminf ?S ≤ ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }  { fix n show "?f n ≤ suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }  show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto  show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by autoqedtheorem summable_Leibniz: fixes a :: "nat => real"  assumes a_zero: "a ----> 0" and "monoseq a"  shows "summable (λ n. (-1)^n * a n)" (is "?summable")  and "0 < a 0 --> (∀n. (∑i. -1^i*a i) ∈ { ∑i=0..<2*n. -1^i * a i .. ∑i=0..<2*n+1. -1^i * a i})" (is "?pos")  and "a 0 < 0 --> (∀n. (∑i. -1^i*a i) ∈ { ∑i=0..<2*n+1. -1^i * a i .. ∑i=0..<2*n. -1^i * a i})" (is "?neg")  and "(λn. ∑i=0..<2*n. -1^i*a i) ----> (∑i. -1^i*a i)" (is "?f")  and "(λn. ∑i=0..<2*n+1. -1^i*a i) ----> (∑i. -1^i*a i)" (is "?g")proof -  have "?summable ∧ ?pos ∧ ?neg ∧ ?f ∧ ?g"  proof (cases "(∀ n. 0 ≤ a n) ∧ (∀m. ∀n≥m. a n ≤ a m)")    case True    hence ord: "!!n m. m ≤ n ==> a n ≤ a m" and ge0: "!! n. 0 ≤ a n" by auto    { fix n have "a (Suc n) ≤ a n" using ord[where n="Suc n" and m=n] by auto }    note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this    from leibniz[OF mono]    show ?thesis using `0 ≤ a 0` by auto  next    let ?a = "λ n. - a n"    case False    with monoseq_le[OF `monoseq a` `a ----> 0`]    have "(∀ n. a n ≤ 0) ∧ (∀m. ∀n≥m. a m ≤ a n)" by auto    hence ord: "!!n m. m ≤ n ==> ?a n ≤ ?a m" and ge0: "!! n. 0 ≤ ?a n" by auto    { fix n have "?a (Suc n) ≤ ?a n" using ord[where n="Suc n" and m=n] by auto }    note monotone = this    note leibniz = summable_Leibniz'[OF _ ge0, of "λx. x", OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]    have "summable (λ n. (-1)^n * ?a n)" using leibniz(1) by auto    then obtain l where "(λ n. (-1)^n * ?a n) sums l" unfolding summable_def by auto    from this[THEN sums_minus]    have "(λ n. (-1)^n * a n) sums -l" by auto    hence ?summable unfolding summable_def by auto    moreover    have "!! a b :: real. ¦ - a - - b ¦ = ¦a - b¦" unfolding minus_diff_minus by auto    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]    have move_minus: "(∑n. - (-1 ^ n * a n)) = - (∑n. -1 ^ n * a n)" by auto    have ?pos using `0 ≤ ?a 0` by auto    moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto    moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] by auto    ultimately show ?thesis by auto  qed  from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]       this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]  show ?summable and ?pos and ?neg and ?f and ?g .qedsubsection {* 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)lemma sums_Suc_imp:  assumes f: "f 0 = 0"  shows "(λn. f (Suc n)) sums s ==> (λn. f n) sums s"unfolding sums_defapply (rule LIMSEQ_imp_Suc)apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])apply (simp only: setsum_shift_bounds_Suc_ivl)donelemma diffs_equiv:  fixes x :: "'a::{real_normed_vector, ring_1}"  shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>      (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums         (∑n. (diffs c)(n) * (x ^ n))"unfolding diffs_defapply (drule summable_sums)apply (rule sums_Suc_imp, simp_all)donelemma lemma_termdiff1:  fixes z :: "'a :: {monoid_mult,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: algebra_simps power_add [symmetric])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 :: {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: power_Suc setsum_op_ivl_Suc of_nat_Suc)apply (subst lemma_realpow_rev_sumr)apply (subst sumr_diff_mult_const2)apply simpapply (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 power_Suc)apply (subst mult_assoc [symmetric], subst power_add [symmetric])apply (simp add: mult_ac)donelemma real_setsum_nat_ivl_bounded2:  fixes K :: "'a::linordered_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)donelemma lemma_termdiff3:  fixes h z :: "'a::{real_normed_field}"  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         of_nat_0_le_iff         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 .qedlemma lemma_termdiff4:  fixes f :: "'a::{real_normed_field} =>              '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"unfolding LIM_eq diff_0_rightproof (safe)  let ?h = "of_real (k / 2)::'a"  have "?h ≠ 0" and "norm ?h < k" using k by simp_all  hence "norm (f ?h) ≤ K * norm ?h" by (rule le)  hence "0 ≤ K * norm ?h" by (rule order_trans [OF norm_ge_zero])  hence zero_le_K: "0 ≤ K" using k by (simp add: zero_le_mult_iff)  fix r::real assume r: "0 < r"  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  qedqedlemma lemma_termdiff5:  fixes g :: "'a::{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" .qedtext{* FIXME: Long proofs*}lemma termdiffs_aux:  fixes x :: "'a::{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  qedqedlemma termdiffs:  fixes K x :: "'a::{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)"unfolding deriv_defproof (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" using 4 by (simp add: less_diff_eq)  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 arg_cong [where f="suminf"], rule ext)      apply (simp add: algebra_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])  qedqedsubsection {* Derivability of power series *}lemma DERIV_series': fixes f :: "real => nat => real"  assumes DERIV_f: "!! n. DERIV (λ x. f x n) x0 :> (f' x0 n)"  and allf_summable: "!! x. x ∈ {a <..< b} ==> summable (f x)" and x0_in_I: "x0 ∈ {a <..< b}"  and "summable (f' x0)"  and "summable L" and L_def: "!! n x y. [| x ∈ { a <..< b} ; y ∈ { a <..< b} |] ==> ¦ f x n - f y n ¦ ≤ L n * ¦ x - y ¦"  shows "DERIV (λ x. suminf (f x)) x0 :> (suminf (f' x0))"  unfolding deriv_defproof (rule LIM_I)  fix r :: real assume "0 < r" hence "0 < r/3" by auto  obtain N_L where N_L: "!! n. N_L ≤ n ==> ¦ ∑ i. L (i + n) ¦ < r/3"    using suminf_exist_split[OF `0 < r/3` `summable L`] by auto  obtain N_f' where N_f': "!! n. N_f' ≤ n ==> ¦ ∑ i. f' x0 (i + n) ¦ < r/3"    using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto  let ?N = "Suc (max N_L N_f')"  have "¦ ∑ i. f' x0 (i + ?N) ¦ < r/3" (is "?f'_part < r/3") and    L_estimate: "¦ ∑ i. L (i + ?N) ¦ < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto  let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"  let ?r = "r / (3 * real ?N)"  have "0 < 3 * real ?N" by auto  from divide_pos_pos[OF `0 < r` this]  have "0 < ?r" .  let "?s n" = "SOME s. 0 < s ∧ (∀ x. x ≠ 0 ∧ ¦ x ¦ < s --> ¦ ?diff n x - f' x0 n ¦ < ?r)"  def S' ≡ "Min (?s ` { 0 ..< ?N })"  have "0 < S'" unfolding S'_def  proof (rule iffD2[OF Min_gr_iff])    show "∀ x ∈ (?s ` { 0 ..< ?N }). 0 < x"    proof (rule ballI)      fix x assume "x ∈ ?s ` {0..<?N}"      then obtain n where "x = ?s n" and "n ∈ {0..<?N}" using image_iff[THEN iffD1] by blast      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]      obtain s where s_bound: "0 < s ∧ (∀x. x ≠ 0 ∧ ¦x¦ < s --> ¦?diff n x - f' x0 n¦ < ?r)" by auto      have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)      thus "0 < x" unfolding `x = ?s n` .    qed  qed auto  def S ≡ "min (min (x0 - a) (b - x0)) S'"  hence "0 < S" and S_a: "S ≤ x0 - a" and S_b: "S ≤ b - x0" and "S ≤ S'" using x0_in_I and `0 < S'`    by auto  { fix x assume "x ≠ 0" and "¦ x ¦ < S"    hence x_in_I: "x0 + x ∈ { a <..< b }" using S_a S_b by auto    note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]    note div_smbl = summable_divide[OF diff_smbl]    note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]    note ign = summable_ignore_initial_segment[where k="?N"]    note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]    note div_shft_smbl = summable_divide[OF diff_shft_smbl]    note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]    { fix n      have "¦ ?diff (n + ?N) x ¦ ≤ L (n + ?N) * ¦ (x0 + x) - x0 ¦ / ¦ x ¦"        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .      hence "¦ ( ¦ ?diff (n + ?N) x ¦) ¦ ≤ L (n + ?N)" using `x ≠ 0` by auto    } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]    from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]    have "¦ ∑ i. ?diff (i + ?N) x ¦ ≤ (∑ i. L (i + ?N))" .    hence "¦ ∑ i. ?diff (i + ?N) x ¦ ≤ r / 3" (is "?L_part ≤ r/3") using L_estimate by auto    have "¦∑n ∈ { 0 ..< ?N}. ?diff n x - f' x0 n ¦ ≤ (∑n ∈ { 0 ..< ?N}. ¦?diff n x - f' x0 n ¦)" ..    also have "… < (∑n ∈ { 0 ..< ?N}. ?r)"    proof (rule setsum_strict_mono)      fix n assume "n ∈ { 0 ..< ?N}"      have "¦ x ¦ < S" using `¦ x ¦ < S` .      also have "S ≤ S'" using `S ≤ S'` .      also have "S' ≤ ?s n" unfolding S'_def      proof (rule Min_le_iff[THEN iffD2])        have "?s n ∈ (?s ` {0..<?N}) ∧ ?s n ≤ ?s n" using `n ∈ { 0 ..< ?N}` by auto        thus "∃ a ∈ (?s ` {0..<?N}). a ≤ ?s n" by blast      qed auto      finally have "¦ x ¦ < ?s n" .      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]      have "∀x. x ≠ 0 ∧ ¦x¦ < ?s n --> ¦?diff n x - f' x0 n¦ < ?r" .      with `x ≠ 0` and `¦x¦ < ?s n`      show "¦?diff n x - f' x0 n¦ < ?r" by blast    qed auto    also have "… = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)    also have "… = real ?N * ?r" unfolding real_eq_of_nat by auto    also have "… = r/3" by auto    finally have "¦∑n ∈ { 0 ..< ?N}. ?diff n x - f' x0 n ¦ < r / 3" (is "?diff_part < r / 3") .    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]    have "¦ (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) ¦ =                    ¦ ∑n. ?diff n x - f' x0 n ¦" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto    also have "… ≤ ?diff_part + ¦ (∑n. ?diff (n + ?N) x) - (∑ n. f' x0 (n + ?N)) ¦" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)    also have "… ≤ ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto    also have "… < r /3 + r/3 + r/3"      using `?diff_part < r/3` `?L_part ≤ r/3` and `?f'_part < r/3`      by (rule add_strict_mono [OF add_less_le_mono])    finally have "¦ (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) ¦ < r"      by auto  } thus "∃ s > 0. ∀ x. x ≠ 0 ∧ norm (x - 0) < s -->      norm (((∑n. f (x0 + x) n) - (∑n. f x0 n)) / x - (∑n. f' x0 n)) < r" using `0 < S`    unfolding real_norm_def diff_0_right by blastqedlemma DERIV_power_series': fixes f :: "nat => real"  assumes converges: "!! x. x ∈ {-R <..< R} ==> summable (λ n. f n * real (Suc n) * x^n)"  and x0_in_I: "x0 ∈ {-R <..< R}" and "0 < R"  shows "DERIV (λ x. (∑ n. f n * x^(Suc n))) x0 :> (∑ n. f n * real (Suc n) * x0^n)"  (is "DERIV (λ x. (suminf (?f x))) x0 :> (suminf (?f' x0))")proof -  { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"    hence "x0 ∈ {-R' <..< R'}" and "R' ∈ {-R <..< R}" and "x0 ∈ {-R <..< R}" by auto    have "DERIV (λ x. (suminf (?f x))) x0 :> (suminf (?f' x0))"    proof (rule DERIV_series')      show "summable (λ n. ¦f n * real (Suc n) * R'^n¦)"      proof -        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto        hence in_Rball: "(R' + R) / 2 ∈ {-R <..< R}" using `R' < R` by auto        have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto      qed      { fix n x y assume "x ∈ {-R' <..< R'}" and "y ∈ {-R' <..< R'}"        show "¦?f x n - ?f y n¦ ≤ ¦f n * real (Suc n) * R'^n¦ * ¦x-y¦"        proof -          have "¦f n * x ^ (Suc n) - f n * y ^ (Suc n)¦ = (¦f n¦ * ¦x-y¦) * ¦∑p = 0..<Suc n. x ^ p * y ^ (n - p)¦"            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto          also have "… ≤ (¦f n¦ * ¦x-y¦) * (¦real (Suc n)¦ * ¦R' ^ n¦)"          proof (rule mult_left_mono)            have "¦∑p = 0..<Suc n. x ^ p * y ^ (n - p)¦ ≤ (∑p = 0..<Suc n. ¦x ^ p * y ^ (n - p)¦)" by (rule setsum_abs)            also have "… ≤ (∑p = 0..<Suc n. R' ^ n)"            proof (rule setsum_mono)              fix p assume "p ∈ {0..<Suc n}" hence "p ≤ n" by auto              { fix n fix x :: real assume "x ∈ {-R'<..<R'}"                hence "¦x¦ ≤ R'"  by auto                hence "¦x^n¦ ≤ R'^n" unfolding power_abs by (rule power_mono, auto)              } from mult_mono[OF this[OF `x ∈ {-R'<..<R'}`, of p] this[OF `y ∈ {-R'<..<R'}`, of "n-p"]] `0 < R'`              have "¦x^p * y^(n-p)¦ ≤ R'^p * R'^(n-p)" unfolding abs_mult by auto              thus "¦x^p * y^(n-p)¦ ≤ R'^n" unfolding power_add[symmetric] using `p ≤ n` by auto            qed            also have "… = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto            finally show "¦∑p = 0..<Suc n. x ^ p * y ^ (n - p)¦ ≤ ¦real (Suc n)¦ * ¦R' ^ n¦" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .            show "0 ≤ ¦f n¦ * ¦x - y¦" unfolding abs_mult[symmetric] by auto          qed          also have "… = ¦f n * real (Suc n) * R' ^ n¦ * ¦x - y¦" unfolding abs_mult mult_assoc[symmetric] by algebra          finally show ?thesis .        qed }      { fix n show "DERIV (λ x. ?f x n) x0 :> (?f' x0 n)"          by (auto intro!: DERIV_intros simp del: power_Suc) }      { fix x assume "x ∈ {-R' <..< R'}" hence "R' ∈ {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto        have "summable (λ n. f n * x^n)"        proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' ∈ {-R <..< R}`] `norm x < norm R'`]], rule allI)          fix n          have le: "¦f n¦ * 1 ≤ ¦f n¦ * real (Suc n)" by (rule mult_left_mono, auto)          show "¦f n * x ^ n¦ ≤ norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult            by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])        qed        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]        show "summable (?f x)" by auto }      show "summable (?f' x0)" using converges[OF `x0 ∈ {-R <..< R}`] .      show "x0 ∈ {-R' <..< R'}" using `x0 ∈ {-R' <..< R'}` .    qed  } note for_subinterval = this  let ?R = "(R + ¦x0¦) / 2"  have "¦x0¦ < ?R" using assms by auto  hence "- ?R < x0"  proof (cases "x0 < 0")    case True    hence "- x0 < ?R" using `¦x0¦ < ?R` by auto    thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto  next    case False    have "- ?R < 0" using assms by auto    also have "… ≤ x0" using False by auto    finally show ?thesis .  qed  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto  from for_subinterval[OF this]  show ?thesis .qedsubsection {* Exponential Function *}definition exp :: "'a => 'a::{real_normed_field,banach}" where  "exp = (λx. ∑n. x ^ n /⇩R real (fact n))"lemma summable_exp_generic:  fixes x :: "'a::{real_normed_algebra_1,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 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 inverse_eq_divide)  qedqedlemma summable_norm_exp:  fixes x :: "'a::{real_normed_algebra_1,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_power_ineq)qedlemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"by (insert summable_exp_generic [where x=x], simp)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 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 DERIV_exp [simp]: "DERIV exp x :> exp(x)"unfolding exp_def scaleR_conv_of_realapply (rule DERIV_cong)apply (rule termdiffs [where K="of_real (1 + norm x)"])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)donelemma isCont_exp: "isCont exp x"  by (rule DERIV_exp [THEN DERIV_isCont])lemma isCont_exp' [simp]: "isCont f a ==> isCont (λx. exp (f x)) a"  by (rule isCont_o2 [OF _ isCont_exp])lemma tendsto_exp [tendsto_intros]:  "(f ---> a) F ==> ((λx. exp (f x)) ---> exp a) F"  by (rule isCont_tendsto_compose [OF isCont_exp])subsubsection {* Properties of the Exponential Function *}lemma powser_zero:  fixes f :: "nat => 'a::{real_normed_algebra_1}"  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 unfolding One_nat_def by simpqedlemma exp_zero [simp]: "exp 0 = 1"unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)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_head_Suc setsum_shift_bounds_cl_Suc_ivl         del: setsum_cl_ivl_Suc)lemma exp_series_add:  fixes x y :: "'a::{real_field}"  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 simpnext  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 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 distrib_right)  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 del: setsum_cl_ivl_Suc)qedlemma exp_add: "exp (x + y) = exp x * exp y"unfolding exp_defby (simp only: Cauchy_product summable_norm_exp exp_series_add)lemma mult_exp_exp: "exp x * exp y = exp (x + y)"by (rule exp_add [symmetric])lemma exp_of_real: "exp (of_real x) = of_real (exp x)"unfolding exp_defapply (subst suminf_of_real)apply (rule summable_exp_generic)apply (simp add: scaleR_conv_of_real)donelemma exp_not_eq_zero [simp]: "exp x ≠ 0"proof  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)  also assume "exp x = 0"  finally show "False" by simpqedlemma exp_minus: "exp (- x) = inverse (exp x)"by (rule inverse_unique [symmetric], simp add: mult_exp_exp)lemma exp_diff: "exp (x - y) = exp x / exp y"  unfolding diff_minus divide_inverse  by (simp add: exp_add exp_minus)subsubsection {* Properties of the Exponential Function on Reals *}text {* Comparisons of @{term "exp x"} with zero. *}text{*Proof: because every exponential can be seen as a square.*}lemma exp_ge_zero [simp]: "0 ≤ exp (x::real)"proof -  have "0 ≤ exp (x/2) * exp (x/2)" by simp  thus ?thesis by (simp add: exp_add [symmetric])qedlemma exp_gt_zero [simp]: "0 < exp (x::real)"by (simp add: order_less_le)lemma not_exp_less_zero [simp]: "¬ exp (x::real) < 0"by (simp add: not_less)lemma not_exp_le_zero [simp]: "¬ exp (x::real) ≤ 0"by (simp add: not_le)lemma abs_exp_cancel [simp]: "¦exp x::real¦ = exp x"by simplemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"apply (induct "n")apply (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)donetext {* Strict monotonicity of exponential. *}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 order_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_mult_iff)donelemma exp_gt_one: "0 < (x::real) ==> 1 < exp x"proof -  assume x: "0 < x"  hence "1 < 1 + x" by simp  also from x have "1 + x ≤ exp x"    by (simp add: exp_ge_add_one_self_aux)  finally show ?thesis .qedlemma exp_less_mono:  fixes x y :: real  assumes "x < y" shows "exp x < exp y"proof -  from `x < y` have "0 < y - x" by simp  hence "1 < exp (y - x)" by (rule exp_gt_one)  hence "1 < exp y / exp x" by (simp only: exp_diff)  thus "exp x < exp y" by simpqedlemma 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)donelemma 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)text {* Comparisons of @{term "exp x"} with one. *}lemma one_less_exp_iff [simp]: "1 < exp (x::real) <-> 0 < x"  using exp_less_cancel_iff [where x=0 and y=x] by simplemma exp_less_one_iff [simp]: "exp (x::real) < 1 <-> x < 0"  using exp_less_cancel_iff [where x=x and y=0] by simplemma one_le_exp_iff [simp]: "1 ≤ exp (x::real) <-> 0 ≤ x"  using exp_le_cancel_iff [where x=0 and y=x] by simplemma exp_le_one_iff [simp]: "exp (x::real) ≤ 1 <-> x ≤ 0"  using exp_le_cancel_iff [where x=x and y=0] by simplemma exp_eq_one_iff [simp]: "exp (x::real) = 1 <-> x = 0"  using exp_inj_iff [where x=x and y=0] by simplemma lemma_exp_total: "1 ≤ y ==> ∃x. 0 ≤ x & x ≤ y - 1 & exp(x::real) = y"proof (rule IVT)  assume "1 ≤ y"  hence "0 ≤ y - 1" by simp  hence "1 + (y - 1) ≤ exp (y - 1)" by (rule exp_ge_add_one_self_aux)  thus "y ≤ exp (y - 1)" by simpqed (simp_all add: le_diff_eq)lemma exp_total: "0 < (y::real) ==> ∃x. exp x = y"proof (rule linorder_le_cases [of 1 y])  assume "1 ≤ y" thus "∃x. exp x = y"    by (fast dest: lemma_exp_total)next  assume "0 < y" and "y ≤ 1"  hence "1 ≤ inverse y" by (simp add: one_le_inverse_iff)  then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)  hence "exp (- x) = y" by (simp add: exp_minus)  thus "∃x. exp x = y" ..qedsubsection {* Natural Logarithm *}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"  by (metis exp_gt_zero exp_ln)lemma ln_unique: "exp y = x ==> ln x = y"  by (erule subst, rule ln_exp)lemma ln_one [simp]: "ln 1 = 0"  by (rule ln_unique, simp)lemma ln_mult: "[|0 < x; 0 < y|] ==> ln (x * y) = ln x + ln y"  by (rule ln_unique, simp add: exp_add)lemma ln_inverse: "0 < x ==> ln (inverse x) = - ln x"  by (rule ln_unique, simp add: exp_minus)lemma ln_div: "[|0 < x; 0 < y|] ==> ln (x / y) = ln x - ln y"  by (rule ln_unique, simp add: exp_diff)lemma ln_realpow: "0 < x ==> ln (x ^ n) = real n * ln x"  by (rule ln_unique, simp add: exp_real_of_nat_mult)lemma ln_less_cancel_iff [simp]: "[|0 < x; 0 < y|] ==> ln x < ln y <-> x < y"  by (subst exp_less_cancel_iff [symmetric], simp)lemma ln_le_cancel_iff [simp]: "[|0 < x; 0 < y|] ==> ln x ≤ ln y <-> x ≤ y"  by (simp add: linorder_not_less [symmetric])lemma ln_inj_iff [simp]: "[|0 < x; 0 < y|] ==> ln x = ln y <-> x = y"  by (simp add: order_eq_iff)lemma ln_add_one_self_le_self [simp]: "0 ≤ x ==> ln (1 + x) ≤ x"  apply (rule exp_le_cancel_iff [THEN iffD1])  apply (simp add: exp_ge_add_one_self_aux)  donelemma ln_less_self [simp]: "0 < x ==> ln x < x"  by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_alllemma ln_ge_zero [simp]: "1 ≤ x ==> 0 ≤ ln x"  using ln_le_cancel_iff [of 1 x] by simplemma ln_ge_zero_imp_ge_one: "[|0 ≤ ln x; 0 < x|] ==> 1 ≤ x"  using ln_le_cancel_iff [of 1 x] by simplemma ln_ge_zero_iff [simp]: "0 < x ==> (0 ≤ ln x) = (1 ≤ x)"  using ln_le_cancel_iff [of 1 x] by simplemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"  using ln_less_cancel_iff [of x 1] by simplemma ln_gt_zero: "1 < x ==> 0 < ln x"  using ln_less_cancel_iff [of 1 x] by simplemma ln_gt_zero_imp_gt_one: "[|0 < ln x; 0 < x|] ==> 1 < x"  using ln_less_cancel_iff [of 1 x] by simplemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"  using ln_less_cancel_iff [of 1 x] by simplemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"  using ln_inj_iff [of x 1] by simplemma ln_less_zero: "[|0 < x; x < 1|] ==> ln x < 0"  by simplemma 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)  donelemma tendsto_ln [tendsto_intros]:  "[|(f ---> a) F; 0 < a|] ==> ((λx. ln (f x)) ---> ln a) F"  by (rule isCont_tendsto_compose [OF isCont_ln])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 DERIV_cong [OF DERIV_exp exp_ln])  apply (simp_all add: abs_if isCont_ln)  donelemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"  by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)lemma ln_series: assumes "0 < x" and "x < 2"  shows "ln x = (∑ n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")proof -  let "?f' x n" = "(-1)^n * (x - 1)^n"  have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"  proof (rule DERIV_isconst3[where x=x])    fix x :: real assume "x ∈ {0 <..< 2}" hence "0 < x" and "x < 2" by auto    have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto    have "1 / x = 1 / (1 - (1 - x))" by auto    also have "… = (∑ n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)    also have "… = suminf (?f' x)" unfolding power_mult_distrib[symmetric] by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)    finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto    moreover    have repos: "!! h x :: real. h - 1 + x = h + x - 1" by auto    have "DERIV (λx. suminf (?f x)) (x - 1) :> (∑n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"    proof (rule DERIV_power_series')      show "x - 1 ∈ {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto      { fix x :: real assume "x ∈ {- 1<..<1}" hence "norm (-x) < 1" by auto        show "summable (λn. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"          unfolding One_nat_def          by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])      }    qed    hence "DERIV (λx. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto    hence "DERIV (λx. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .    ultimately have "DERIV (λx. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"      by (rule DERIV_diff)    thus "DERIV (λx. ln x - suminf (?f (x - 1))) x :> 0" by auto  qed (auto simp add: assms)  thus ?thesis by autoqedlemma exp_first_two_terms: "exp x = 1 + x + (∑ n. inverse(fact (n+2)) * (x ^ (n+2)))"proof -  have "exp x = suminf (%n. inverse(fact n) * (x ^ n))"    by (simp add: exp_def)  also from summable_exp have "... = (∑ n::nat = 0 ..< 2. inverse(fact n) * (x ^ n)) +      (∑ n. inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")    by (rule suminf_split_initial_segment)  also have "?a = 1 + x"    by (simp add: numeral_2_eq_2)  finally show ?thesis .qedlemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x^2"proof -  assume a: "0 <= x"  assume b: "x <= 1"  { fix n :: nat    have "2 * 2 ^ n ≤ fact (n + 2)"      by (induct n, simp, simp)    hence "real ((2::nat) * 2 ^ n) ≤ real (fact (n + 2))"      by (simp only: real_of_nat_le_iff)    hence "2 * 2 ^ n ≤ real (fact (n + 2))"      by simp    hence "inverse (fact (n + 2)) ≤ inverse (2 * 2 ^ n)"      by (rule le_imp_inverse_le) simp    hence "inverse (fact (n + 2)) ≤ 1/2 * (1/2)^n"      by (simp add: inverse_mult_distrib power_inverse)    hence "inverse (fact (n + 2)) * (x^n * x²) ≤ 1/2 * (1/2)^n * (1 * x²)"      by (rule mult_mono)        (rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)    hence "inverse (fact (n + 2)) * x ^ (n + 2) ≤ (x²/2) * ((1/2)^n)"      unfolding power_add by (simp add: mult_ac del: fact_Suc) }  note aux1 = this  have "(λn. x² / 2 * (1 / 2) ^ n) sums (x² / 2 * (1 / (1 - 1 / 2)))"    by (intro sums_mult geometric_sums, simp)  hence aux2: "(λn. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2"    by simp  have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <= x^2"  proof -    have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=        suminf (%n. (x^2/2) * ((1/2)^n))"      apply (rule summable_le)      apply (rule allI, rule aux1)      apply (rule summable_exp [THEN summable_ignore_initial_segment])      by (rule sums_summable, rule aux2)    also have "... = x^2"      by (rule sums_unique [THEN sym], rule aux2)    finally show ?thesis .  qed  thus ?thesis unfolding exp_first_two_terms by autoqedlemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"proof -  assume a: "0 <= (x::real)" and b: "x < 1"  have "(1 - x) * (1 + x + x^2) = (1 - x^3)"    by (simp add: algebra_simps power2_eq_square power3_eq_cube)  also have "... <= 1"    by (auto simp add: a)  finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .  moreover have c: "0 < 1 + x + x²"    by (simp add: add_pos_nonneg a)  ultimately have "1 - x <= 1 / (1 + x + x^2)"    by (elim mult_imp_le_div_pos)  also have "... <= 1 / exp x"    apply (rule divide_left_mono)    apply (rule exp_bound, rule a)    apply (rule b [THEN less_imp_le])    apply simp    apply (rule mult_pos_pos)    apply (rule c)    apply simp    done  also have "... = exp (-x)"    by (auto simp add: exp_minus divide_inverse)  finally have "1 - x <= exp (- x)" .  also have "1 - x = exp (ln (1 - x))"  proof -    have "0 < 1 - x"      by (insert b, auto)    thus ?thesis      by (auto simp only: exp_ln_iff [THEN sym])  qed  finally have "exp (ln (1 - x)) <= exp (- x)" .  thus ?thesis by (auto simp only: exp_le_cancel_iff)qedlemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"  apply (case_tac "0 <= x")  apply (erule exp_ge_add_one_self_aux)  apply (case_tac "x <= -1")  apply (subgoal_tac "1 + x <= 0")  apply (erule order_trans)  apply simp  apply simp  apply (subgoal_tac "1 + x = exp(ln (1 + x))")  apply (erule ssubst)  apply (subst exp_le_cancel_iff)  apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")  apply simp  apply (rule ln_one_minus_pos_upper_bound)  apply autodonelemma exp_at_bot: "(exp ---> (0::real)) at_bot"  unfolding tendsto_Zfun_iffproof (rule ZfunI, simp add: eventually_at_bot_dense)  fix r :: real assume "0 < r"  { fix x assume "x < ln r"    then have "exp x < exp (ln r)"      by simp    with `0 < r` have "exp x < r"      by simp }  then show "∃k. ∀n<k. exp n < r" by autoqedlemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"  by (rule filterlim_at_top_at_top[where Q="λx. True" and P="λx. 0 < x" and g="ln"])     (auto intro: eventually_gt_at_top)lemma ln_at_0: "LIM x at_right 0. ln x :> at_bot"  by (rule filterlim_at_bot_at_right[where Q="λx. 0 < x" and P="λx. True" and g="exp"])     (auto simp: eventually_within)lemma ln_at_top: "LIM x at_top. ln x :> at_top"  by (rule filterlim_at_top_at_top[where Q="λx. 0 < x" and P="λx. True" and g="exp"])     (auto intro: eventually_gt_at_top)lemma tendsto_power_div_exp_0: "((λx. x ^ k / exp x) ---> (0::real)) at_top"proof (induct k)  show "((λx. x ^ 0 / exp x) ---> (0::real)) at_top"    by (simp add: inverse_eq_divide[symmetric])       (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono              at_top_le_at_infinity order_refl)next  case (Suc k)  show ?case  proof (rule lhospital_at_top_at_top)    show "eventually (λx. DERIV (λx. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"      by eventually_elim (intro DERIV_intros, simp, simp)    show "eventually (λx. DERIV exp x :> exp x) at_top"      by eventually_elim (auto intro!: DERIV_intros)    show "eventually (λx. exp x ≠ 0) at_top"      by auto    from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]    show "((λx. real (Suc k) * x ^ k / exp x) ---> 0) at_top"      by simp  qed (rule exp_at_top)qedsubsection {* Sine and Cosine *}definition sin_coeff :: "nat => real" where  "sin_coeff = (λn. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"definition cos_coeff :: "nat => real" where  "cos_coeff = (λn. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"definition sin :: "real => real" where  "sin = (λx. ∑n. sin_coeff n * x ^ n)"definition cos :: "real => real" where  "cos = (λx. ∑n. cos_coeff n * x ^ n)"lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"  unfolding sin_coeff_def by simplemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"  unfolding cos_coeff_def by simplemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"  unfolding cos_coeff_def sin_coeff_def  by (simp del: mult_Suc)lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"  unfolding cos_coeff_def sin_coeff_def  by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)lemma summable_sin: "summable (λn. sin_coeff n * x ^ n)"unfolding sin_coeff_defapply (rule summable_comparison_test [OF _ summable_exp [where x="¦x¦"]])apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)donelemma summable_cos: "summable (λn. cos_coeff n * x ^ n)"unfolding cos_coeff_defapply (rule summable_comparison_test [OF _ summable_exp [where x="¦x¦"]])apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)donelemma sin_converges: "(λn. sin_coeff n * x ^ n) sums sin(x)"unfolding sin_def by (rule summable_sin [THEN summable_sums])lemma cos_converges: "(λn. cos_coeff n * x ^ n) sums cos(x)"unfolding cos_def by (rule summable_cos [THEN summable_sums])lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"  by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)lemma diffs_cos_coeff: "diffs cos_coeff = (λn. - sin_coeff n)"  by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)text{*Now at last we can get the derivatives of exp, sin and cos*}lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"  unfolding sin_def cos_def  apply (rule DERIV_cong, rule termdiffs [where K="1 + ¦x¦"])  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff    summable_minus summable_sin summable_cos)  donelemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"  unfolding cos_def sin_def  apply (rule DERIV_cong, rule termdiffs [where K="1 + ¦x¦"])  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus    summable_minus summable_sin summable_cos suminf_minus)  donelemma isCont_sin: "isCont sin x"  by (rule DERIV_sin [THEN DERIV_isCont])lemma isCont_cos: "isCont cos x"  by (rule DERIV_cos [THEN DERIV_isCont])lemma isCont_sin' [simp]: "isCont f a ==> isCont (λx. sin (f x)) a"  by (rule isCont_o2 [OF _ isCont_sin])lemma isCont_cos' [simp]: "isCont f a ==> isCont (λx. cos (f x)) a"  by (rule isCont_o2 [OF _ isCont_cos])lemma tendsto_sin [tendsto_intros]:  "(f ---> a) F ==> ((λx. sin (f x)) ---> sin a) F"  by (rule isCont_tendsto_compose [OF isCont_sin])lemma tendsto_cos [tendsto_intros]:  "(f ---> a) F ==> ((λx. cos (f x)) ---> cos a) F"  by (rule isCont_tendsto_compose [OF isCont_cos])declare  DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]  DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]  DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]  DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]subsection {* Properties of Sine and Cosine *}lemma sin_zero [simp]: "sin 0 = 0"  unfolding sin_def sin_coeff_def by (simp add: powser_zero)lemma cos_zero [simp]: "cos 0 = 1"  unfolding cos_def cos_coeff_def by (simp add: powser_zero)lemma sin_cos_squared_add [simp]: "(sin x)² + (cos x)² = 1"proof -  have "∀x. DERIV (λx. (sin x)² + (cos x)²) x :> 0"    by (auto intro!: DERIV_intros)  hence "(sin x)² + (cos x)² = (sin 0)² + (cos 0)²"    by (rule DERIV_isconst_all)  thus "(sin x)² + (cos x)² = 1" by simpqedlemma sin_cos_squared_add2 [simp]: "(cos x)² + (sin x)² = 1"  by (subst add_commute, rule sin_cos_squared_add)lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"  using sin_cos_squared_add2 [unfolded power2_eq_square] .lemma sin_squared_eq: "(sin x)² = 1 - (cos x)²"  unfolding eq_diff_eq by (rule sin_cos_squared_add)lemma cos_squared_eq: "(cos x)² = 1 - (sin x)²"  unfolding eq_diff_eq by (rule sin_cos_squared_add2)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"  using abs_sin_le_one [of x] unfolding abs_le_iff by simplemma sin_le_one [simp]: "sin x ≤ 1"  using abs_sin_le_one [of x] unfolding abs_le_iff by simplemma 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"  using abs_cos_le_one [of x] unfolding abs_le_iff by simplemma cos_le_one [simp]: "cos x ≤ 1"  using abs_cos_le_one [of x] unfolding abs_le_iff by simplemma DERIV_fun_pow: "DERIV g x :> m ==>      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"  by (auto intro!: DERIV_intros)lemma DERIV_fun_exp:     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"  by (auto intro!: DERIV_intros)lemma DERIV_fun_sin:     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"  by (auto intro!: DERIV_intros)lemma DERIV_fun_cos:     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"  by (auto intro!: DERIV_intros)lemma sin_cos_add_lemma:     "(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"  (is "?f x = 0")proof -  have "∀x. DERIV (λx. ?f x) x :> 0"    by (auto intro!: DERIV_intros simp add: algebra_simps)  hence "?f x = ?f 0"    by (rule DERIV_isconst_all)  thus ?thesis by simpqedlemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"  using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simplemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"  using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simplemma sin_cos_minus_lemma:  "(sin(-x) + sin(x))² + (cos(-x) - cos(x))² = 0" (is "?f x = 0")proof -  have "∀x. DERIV (λx. ?f x) x :> 0"    by (auto intro!: DERIV_intros simp add: algebra_simps)  hence "?f x = ?f 0"    by (rule DERIV_isconst_all)  thus ?thesis by simpqedlemma sin_minus [simp]: "sin (-x) = -sin(x)"  using sin_cos_minus_lemma [where x=x] by simplemma cos_minus [simp]: "cos (-x) = cos(x)"  using sin_cos_minus_lemma [where x=x] by simplemma 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"  using sin_add [where x=x and y=x] by simplemma cos_double: "cos(2* x) = ((cos x)²) - ((sin x)²)"  using cos_add [where x=x and y=x]  by (simp add: power2_eq_square)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. sin_coeff k * x ^ k) sums sin x"    by (rule sin_converges [THEN sums_group], simp)  thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)qedlemma sin_gt_zero:  assumes "0 < x" and "x < 2" shows "0 < sin x"proof -  let ?f = "λn. ∑k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"  have pos: "∀n. 0 < ?f n"  proof    fix n :: nat    let ?k2 = "real (Suc (Suc (4 * n)))"    let ?k3 = "real (Suc (Suc (Suc (4 * n))))"    have "x * x < ?k2 * ?k3"      using assms by (intro mult_strict_mono', simp_all)    hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"      by (intro mult_strict_right_mono zero_less_power `0 < x`)    thus "0 < ?f n"      by (simp del: mult_Suc,        simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)  qed  have sums: "?f sums sin x"    by (rule sin_paired [THEN sums_group], simp)  show "0 < sin x"    unfolding sums_unique [OF sums]    using sums_summable [OF sums] pos    by (rule suminf_gt_zero)qedlemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"apply (cut_tac x = x in sin_gt_zero)apply (auto simp add: cos_squared_eq cos_double)donelemma cos_paired:     "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"proof -  have "(λn. ∑k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"    by (rule cos_converges [THEN sums_group], simp)  thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)qedlemma real_mult_inverse_cancel:     "[|(0::real) < x; 0 < x1; x1 * y < x * u |]      ==> inverse x * y < inverse x1 * u"apply (rule_tac c=x in mult_less_imp_less_left)apply (auto simp add: mult_assoc [symmetric])apply (simp (no_asm) add: mult_ac)apply (rule_tac c=x1 in mult_less_imp_less_right)apply (auto simp add: mult_ac)donelemma real_mult_inverse_cancel2:     "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)donelemma realpow_num_eq_if:  fixes m :: "'a::power"  shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"by (cases n, auto)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_nat realpow_num_eq_if del: fact_Suc)apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)apply (rule sumr_pos_lt_pair)apply (erule sums_summable, safe)unfolding One_nat_defapply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]            del: fact_Suc)apply (simp add: inverse_eq_divide less_divide_eq del: fact_Suc)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_ge_zero) prefer 2 apply forceapply (rule real_of_nat_less_iff [THEN iffD2])apply (rule fact_less_mono_nat, auto)donelemmas 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"proof (rule ex_ex1I)  show "∃x. 0 ≤ x & x ≤ 2 & cos x = 0"    by (rule IVT2, simp_all)next  fix x y  assume x: "0 ≤ x ∧ x ≤ 2 ∧ cos x = 0"  assume y: "0 ≤ y ∧ y ≤ 2 ∧ cos y = 0"  have [simp]: "∀x. cos differentiable x"    unfolding differentiable_def by (auto intro: DERIV_cos)  from x y show "x = y"    apply (cut_tac less_linear [of x y], auto)    apply (drule_tac f = cos in Rolle)    apply (drule_tac [5] f = cos in Rolle)    apply (auto dest!: DERIV_cos [THEN DERIV_unique])    apply (metis order_less_le_trans less_le sin_gt_zero)    apply (metis order_less_le_trans less_le sin_gt_zero)    doneqedlemma 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)donelemmas 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)donelemmas 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: "-(pi/2) < 0"by simplemma m2pi_less_pi: "- (2 * pi) < pi"by simplemma 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_1_iff)donelemma 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: cos_diff)lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"by (simp add: cos_add)lemma cos_sin_eq: "cos x = sin (pi/2 - x)"by (simp add: sin_diff)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 distrib_right)donelemma 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 .qedlemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"apply (induct "n")apply (auto simp add: real_of_nat_Suc distrib_right)donelemma 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 simplemma 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)donelemma sin_less_zero:  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"proof -  have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)  thus ?thesis by simpqedlemma 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)donelemma 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)donelemma 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)donelemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"by (simp add: sin_cos_eq cos_gt_zero_pi)lemma pi_ge_two: "2 ≤ pi"proof (rule ccontr)  assume "¬ 2 ≤ pi" hence "pi < 2" by auto  have "∃y > pi. y < 2 ∧ y < 2 * pi"  proof (cases "2 < 2 * pi")    case True with dense[OF `pi < 2`] show ?thesis by auto  next    case False have "pi < 2 * pi" by auto    from dense[OF this] and False show ?thesis by auto  qed  then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast  hence "0 < sin y" using sin_gt_zero by auto  moreover  have "sin y < 0" using sin_gt_zero_pi[of "y - pi"] `pi < y` and `y < 2 * pi` sin_periodic_pi[of "y - pi"] by auto  ultimately show False by autoqedlemma sin_ge_zero: "[| 0 ≤ x; x ≤ pi |] ==> 0 ≤ sin x"by (auto simp add: order_le_less sin_gt_zero_pi)text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.  It should be possible to factor out some of the common parts. *}lemma cos_total: "[| -1 ≤ y; y ≤ 1 |] ==> EX! x. 0 ≤ x & x ≤ pi & (cos x = y)"proof (rule ex_ex1I)  assume y: "-1 ≤ y" "y ≤ 1"  show "∃x. 0 ≤ x & x ≤ pi & cos x = y"    by (rule IVT2, simp_all add: y)next  fix a b  assume a: "0 ≤ a ∧ a ≤ pi ∧ cos a = y"  assume b: "0 ≤ b ∧ b ≤ pi ∧ cos b = y"  have [simp]: "∀x. cos differentiable x"    unfolding differentiable_def by (auto intro: DERIV_cos)  from a b show "a = b"    apply (cut_tac less_linear [of a b], auto)    apply (drule_tac f = cos in Rolle)    apply (drule_tac [5] f = cos in Rolle)    apply (auto dest!: DERIV_cos [THEN DERIV_unique])    apply (metis order_less_le_trans less_le sin_gt_zero_pi)    apply (metis order_less_le_trans less_le sin_gt_zero_pi)    doneqedlemma 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 simpapply (cut_tac y="-y" in cos_total, simp) apply simpapply (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 add: cos_add)donelemma 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: algebra_simps real_of_nat_Suc) prefer 2 apply (simp add: cos_diff)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 algebra_simps, auto)donelemma 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 distrib_right)apply (rule cos_zero_lemma)apply (simp_all add: cos_add)donelemma 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 distrib_right)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)donelemma cos_monotone_0_pi: assumes "0 ≤ y" and "y < x" and "x ≤ pi"  shows "cos x < cos y"proof -  have "- (x - y) < 0" using assms by auto  from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]  obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto  hence "0 < z" and "z < pi" using assms by auto  hence "0 < sin z" using sin_gt_zero_pi by auto  hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)  thus ?thesis by autoqedlemma cos_monotone_0_pi': assumes "0 ≤ y" and "y ≤ x" and "x ≤ pi" shows "cos x ≤ cos y"proof (cases "y < x")  case True show ?thesis using cos_monotone_0_pi[OF `0 ≤ y` True `x ≤ pi`] by autonext  case False hence "y = x" using `y ≤ x` by auto  thus ?thesis by autoqedlemma cos_monotone_minus_pi_0: assumes "-pi ≤ y" and "y < x" and "x ≤ 0"  shows "cos y < cos x"proof -  have "0 ≤ -x" and "-x < -y" and "-y ≤ pi" using assms by auto  from cos_monotone_0_pi[OF this]  show ?thesis unfolding cos_minus .qedlemma cos_monotone_minus_pi_0': assumes "-pi ≤ y" and "y ≤ x" and "x ≤ 0" shows "cos y ≤ cos x"proof (cases "y < x")  case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi ≤ y` True `x ≤ 0`] by autonext  case False hence "y = x" using `y ≤ x` by auto  thus ?thesis by autoqedlemma sin_monotone_2pi': assumes "- (pi / 2) ≤ y" and "y ≤ x" and "x ≤ pi / 2" shows "sin y ≤ sin x"proof -  have "0 ≤ y + pi / 2" and "y + pi / 2 ≤ x + pi / 2" and "x + pi /2 ≤ pi"    using pi_ge_two and assms by auto  from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by autoqedsubsection {* Tangent *}definition tan :: "real => real" where  "tan = (λx. 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)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)"  by (simp add: tan_def cos_add field_simps)lemma add_tan_eq:  "[|cos x ≠ 0; cos y ≠ 0|] ==> tan x + tan y = sin(x + y)/(cos x * cos y)"  by (simp add: tan_def sin_add field_simps)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))"  by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)lemma tan_double:     "[| cos x ≠ 0; cos (2 * x) ≠ 0 |]      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"  using tan_add [of x x] by (simp add: power2_eq_square)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 assms by (simp only: tan_gt_zero)  thus ?thesis by simpqedlemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"  unfolding tan_def sin_double cos_double sin_squared_eq  by (simp add: power2_eq_square)lemma DERIV_tan [simp]: "cos x ≠ 0 ==> DERIV tan x :> inverse ((cos x)²)"  unfolding tan_def  by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)lemma isCont_tan: "cos x ≠ 0 ==> isCont tan x"  by (rule DERIV_tan [THEN DERIV_isCont])lemma isCont_tan' [simp]:  "[|isCont f a; cos (f a) ≠ 0|] ==> isCont (λx. tan (f x)) a"  by (rule isCont_o2 [OF _ isCont_tan])lemma tendsto_tan [tendsto_intros]:  "[|(f ---> a) F; cos a ≠ 0|] ==> ((λx. tan (f x)) ---> tan a) F"  by (rule isCont_tendsto_compose [OF isCont_tan])lemma LIM_cos_div_sin: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"  by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)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_eq)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 sin_diff cos_diff)apply (rule inverse_less_iff_less [THEN iffD1])apply (auto simp add: divide_inverse)apply (rule mult_pos_pos)apply (subgoal_tac [3] "0 < sin e & 0 < cos e")apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)donelemma 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 forceapply (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)donelemma 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 del: exI intro!: exI)donelemma 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 del: exI 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])donelemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"  shows "tan y < tan x"proof -  have "∀ x'. y ≤ x' ∧ x' ≤ x --> DERIV tan x' :> inverse (cos x'^2)"  proof (rule allI, rule impI)    fix x' :: real assume "y ≤ x' ∧ x' ≤ x"    hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto    from cos_gt_zero_pi[OF this]    have "cos x' ≠ 0" by auto    thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)  qed  from MVT2[OF `y < x` this]  obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)²)" by auto  hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto  hence "0 < cos z" using cos_gt_zero_pi by auto  hence inv_pos: "0 < inverse ((cos z)²)" by auto  have "0 < x - y" using `y < x` by auto  from mult_pos_pos [OF this inv_pos]  have "0 < tan x - tan y" unfolding tan_diff by auto  thus ?thesis by autoqedlemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"  shows "(y < x) = (tan y < tan x)"proof  assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by autonext  assume "tan y < tan x"  show "y < x"  proof (rule ccontr)    assume "¬ y < x" hence "x ≤ y" by auto    hence "tan x ≤ tan y"    proof (cases "x = y")      case True thus ?thesis by auto    next      case False hence "x < y" using `x ≤ y` by auto      from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto    qed    thus False using `tan y < tan x` by auto  qedqedlemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by autolemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"  by (simp add: tan_def)lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"proof (induct n arbitrary: x)  case (Suc n)  have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto  show ?case unfolding split_pi_off using Suc by autoqed autolemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"proof (cases "0 ≤ i")  case True hence i_nat: "real i = real (nat i)" by auto  show ?thesis unfolding i_nat by autonext  case False hence i_nat: "real i = - real (nat (-i))" by auto  have "tan x = tan (x + real i * pi - real i * pi)" by auto  also have "… = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)  finally show ?thesis by autoqedlemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"  using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .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)donelemma 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)donelemma 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)donelemma 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)donelemma arccos_cos: "[|0 ≤ x; x ≤ pi |] ==> arccos(cos x) = x"apply (simp add: arccos_def)apply (auto intro!: the1_equality cos_total)donelemma arccos_cos2: "[|x ≤ 0; -pi ≤ x |] ==> arccos(cos x) = -x"apply (simp add: arccos_def)apply (auto intro!: the1_equality cos_total)donelemma 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 simpapply (subgoal_tac "¦x¦² ≤ 1²", simp)apply (rule power_mono, simp, simp)donelemma 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 simpapply (subgoal_tac "¦x¦² ≤ 1²", simp)apply (rule power_mono, simp, simp)donelemma 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 autolemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"by (auto simp only: arctan)lemma arctan_lbound: "- (pi/2) < arctan y"by autolemma arctan_ubound: "arctan y < pi/2"by (auto simp only: arctan)lemma arctan_unique:  assumes "-(pi/2) < x" and "x < pi/2" and "tan x = y"  shows "arctan y = x"  using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)lemma arctan_tan:      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"  by (rule arctan_unique, simp_all)lemma arctan_zero_zero [simp]: "arctan 0 = 0"  by (rule arctan_unique, simp_all)lemma arctan_minus: "arctan (- x) = - arctan x"  apply (rule arctan_unique)  apply (simp only: neg_less_iff_less arctan_ubound)  apply (metis minus_less_iff arctan_lbound)  apply simp  donelemma cos_arctan_not_zero [simp]: "cos (arctan x) ≠ 0"  by (intro less_imp_neq [symmetric] cos_gt_zero_pi    arctan_lbound arctan_ubound)lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x²)"proof (rule power2_eq_imp_eq)  have "0 < 1 + x²" by (simp add: add_pos_nonneg)  show "0 ≤ 1 / sqrt (1 + x²)" by simp  show "0 ≤ cos (arctan x)"    by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)  have "(cos (arctan x))² * (1 + (tan (arctan x))²) = 1"    unfolding tan_def by (simp add: distrib_left power_divide)  thus "(cos (arctan x))² = (1 / sqrt (1 + x²))²"    using `0 < 1 + x²` by (simp add: power_divide eq_divide_eq)qedlemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x²)"  using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]  using tan_arctan [of x] unfolding tan_def cos_arctan  by (simp add: eq_divide_eq)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 distrib_right power_divide tan_def                  mult_assoc power_inverse [symmetric])donelemma arctan_less_iff: "arctan x < arctan y <-> x < y"  by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)lemma arctan_le_iff: "arctan x ≤ arctan y <-> x ≤ y"  by (simp only: not_less [symmetric] arctan_less_iff)lemma arctan_eq_iff: "arctan x = arctan y <-> x = y"  by (simp only: eq_iff [where 'a=real] arctan_le_iff)lemma zero_less_arctan_iff [simp]: "0 < arctan x <-> 0 < x"  using arctan_less_iff [of 0 x] by simplemma arctan_less_zero_iff [simp]: "arctan x < 0 <-> x < 0"  using arctan_less_iff [of x 0] by simplemma zero_le_arctan_iff [simp]: "0 ≤ arctan x <-> 0 ≤ x"  using arctan_le_iff [of 0 x] by simplemma arctan_le_zero_iff [simp]: "arctan x ≤ 0 <-> x ≤ 0"  using arctan_le_iff [of x 0] by simplemma arctan_eq_zero_iff [simp]: "arctan x = 0 <-> x = 0"  using arctan_eq_iff [of x 0] by simplemma 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)donelemma 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)donelemma 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)donelemma 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 (metis arctan_tan order_le_less_trans order_less_le_trans)apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)donelemma 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 DERIV_cong [OF DERIV_sin])apply (simp add: cos_arcsin)apply (subgoal_tac "¦x¦² < 1²", simp)apply (rule power_strict_mono, simp, simp, simp)apply assumptionapply assumptionapply simpapply (erule (1) isCont_arcsin)donelemma 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 DERIV_cong [OF DERIV_cos])apply (simp add: sin_arccos)apply (subgoal_tac "¦x¦² < 1²", simp)apply (rule power_strict_mono, simp, simp, simp)apply assumptionapply assumptionapply simpapply (erule (1) isCont_arccos)donelemma 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 DERIV_cong [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)donedeclare  DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]  DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]  DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"  by (rule filterlim_at_bot_at_right[where Q="λx. - pi/2 < x ∧ x < pi/2" and P="λx. True" and g=arctan])     (auto simp: le_less eventually_within_less dist_real_def simp del: less_divide_eq_numeral1           intro!: tan_monotone exI[of _ "pi/2"])lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"  by (rule filterlim_at_top_at_left[where Q="λx. - pi/2 < x ∧ x < pi/2" and P="λx. True" and g=arctan])     (auto simp: le_less eventually_within_less dist_real_def simp del: less_divide_eq_numeral1           intro!: tan_monotone exI[of _ "pi/2"])lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"proof (rule tendstoI)  fix e :: real assume "0 < e"  def y ≡ "pi/2 - min (pi/2) e"  then have y: "0 ≤ y" "y < pi/2" "pi/2 ≤ e + y"    using `0 < e` by auto  show "eventually (λx. dist (arctan x) (pi / 2) < e) at_top"  proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)    fix x assume "tan y < x"    then have "arctan (tan y) < arctan x"      by (simp add: arctan_less_iff)    with y have "y < arctan x"      by (subst (asm) arctan_tan) simp_all    with arctan_ubound[of x, arith] y `0 < e`    show "dist (arctan x) (pi / 2) < e"      by (simp add: dist_real_def)  qedqedlemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"  unfolding filterlim_at_bot_mirror arctan_minus by (intro tendsto_minus tendsto_arctan_at_top)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 (simp add: cos_ge_zero)  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) simpqedlemma 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: algebra_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) simpqedlemma sin_45: "sin (pi / 4) = sqrt 2 / 2"by (simp add: sin_cos_eq cos_45)lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"by (simp add: sin_cos_eq cos_30)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)donelemma sin_30: "sin (pi / 6) = 1 / 2"by (simp add: sin_cos_eq cos_60)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)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: algebra_simps sin_add)  thus ?thesis    by (simp add: real_of_nat_Suc distrib_right add_divide_distrib                  mult_commute [of pi])qedlemma 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)donelemma 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)donelemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"by (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"  by (auto intro!: DERIV_intros)lemma sin_zero_abs_cos_one: "sin x = 0 ==> ¦cos x¦ = 1"by (auto simp add: sin_zero_iff even_mult_two_ex)lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"by (cut_tac x = x in sin_cos_squared_add3, auto)subsection {* Machins formula *}lemma arctan_one: "arctan 1 = pi / 4"  by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)lemma tan_total_pi4: assumes "¦x¦ < 1"  shows "∃ z. - (pi / 4) < z ∧ z < pi / 4 ∧ tan z = x"proof  show "- (pi / 4) < arctan x ∧ arctan x < pi / 4 ∧ tan (arctan x) = x"    unfolding arctan_one [symmetric] arctan_minus [symmetric]    unfolding arctan_less_iff using assms by autoqedlemma arctan_add: assumes "¦x¦ ≤ 1" and "¦y¦ < 1"  shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"proof (rule arctan_unique [symmetric])  have "- (pi / 4) ≤ arctan x" and "- (pi / 4) < arctan y"    unfolding arctan_one [symmetric] arctan_minus [symmetric]    unfolding arctan_le_iff arctan_less_iff using assms by auto  from add_le_less_mono [OF this]  show 1: "- (pi / 2) < arctan x + arctan y" by simp  have "arctan x ≤ pi / 4" and "arctan y < pi / 4"    unfolding arctan_one [symmetric]    unfolding arctan_le_iff arctan_less_iff using assms by auto  from add_le_less_mono [OF this]  show 2: "arctan x + arctan y < pi / 2" by simp  show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"    using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add)qedtheorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"proof -  have "¦1 / 5¦ < (1 :: real)" by auto  from arctan_add[OF less_imp_le[OF this] this]  have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto  moreover  have "¦5 / 12¦ < (1 :: real)" by auto  from arctan_add[OF less_imp_le[OF this] this]  have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto  moreover  have "¦1¦ ≤ (1::real)" and "¦1 / 239¦ < (1::real)" by auto  from arctan_add[OF this]  have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto  ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto  thus ?thesis unfolding arctan_one by algebraqedsubsection {* Introducing the arcus tangens power series *}lemma monoseq_arctan_series: fixes x :: real  assumes "¦x¦ ≤ 1" shows "monoseq (λ n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")proof (cases "x = 0") case True thus ?thesis unfolding monoseq_def One_nat_def by autonext  case False  have "norm x ≤ 1" and "x ≤ 1" and "-1 ≤ x" using assms by auto  show "monoseq ?a"  proof -    { fix n fix x :: real assume "0 ≤ x" and "x ≤ 1"      have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) ≤ 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"      proof (rule mult_mono)        show "1 / real (Suc (Suc n * 2)) ≤ 1 / real (Suc (n * 2))" by (rule frac_le) simp_all        show "0 ≤ 1 / real (Suc (n * 2))" by auto        show "x ^ Suc (Suc n * 2) ≤ x ^ Suc (n * 2)" by (rule power_decreasing) (simp_all add: `0 ≤ x` `x ≤ 1`)        show "0 ≤ x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 ≤ x`)      qed    } note mono = this    show ?thesis    proof (cases "0 ≤ x")      case True from mono[OF this `x ≤ 1`, THEN allI]      show ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)    next      case False hence "0 ≤ -x" and "-x ≤ 1" using `-1 ≤ x` by auto      from mono[OF this]      have "!!n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) ≥ 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 ≤ -x` by auto      thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])    qed  qedqedlemma zeroseq_arctan_series: fixes x :: real  assumes "¦x¦ ≤ 1" shows "(λ n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")proof (cases "x = 0") case True thus ?thesis unfolding One_nat_def by (auto simp add: tendsto_const)next  case False  have "norm x ≤ 1" and "x ≤ 1" and "-1 ≤ x" using assms by auto  show "?a ----> 0"  proof (cases "¦x¦ < 1")    case True hence "norm x < 1" by auto    from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]    have "(λn. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"      unfolding inverse_eq_divide Suc_eq_plus1 by simp    then show ?thesis using pos2 by (rule LIMSEQ_linear)  next    case False hence "x = -1 ∨ x = 1" using `¦x¦ ≤ 1` by auto    hence n_eq: "!! n. x ^ (n * 2 + 1) = x" unfolding One_nat_def by auto    from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]    show ?thesis unfolding n_eq Suc_eq_plus1 by auto  qedqedlemma summable_arctan_series: fixes x :: real and n :: nat  assumes "¦x¦ ≤ 1" shows "summable (λ k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "summable (?c x)")  by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])lemma less_one_imp_sqr_less_one: fixes x :: real assumes "¦x¦ < 1" shows "x^2 < 1"proof -  from mult_left_mono[OF less_imp_le[OF `¦x¦ < 1`] abs_ge_zero[of x]]  have "¦ x^2 ¦ < 1" using `¦ x ¦ < 1` unfolding numeral_2_eq_2 power_Suc2 by auto  thus ?thesis using zero_le_power2 by autoqedlemma DERIV_arctan_series: assumes "¦ x ¦ < 1"  shows "DERIV (λ x'. ∑ k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (∑ k. (-1)^k * x^(k*2))" (is "DERIV ?arctan _ :> ?Int")proof -  let "?f n" = "if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"  { fix n :: nat assume "even n" hence "2 * (n div 2) = n" by presburger } note n_even=this  have if_eq: "!! n x'. ?f n * real (Suc n) * x'^n = (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" using n_even by auto  { fix x :: real assume "¦x¦ < 1" hence "x^2 < 1" by (rule less_one_imp_sqr_less_one)    have "summable (λ n. -1 ^ n * (x^2) ^n)"      by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x^2 < 1` order_less_imp_le[OF `x^2 < 1`])    hence "summable (λ n. -1 ^ n * x^(2*n))" unfolding power_mult .  } note summable_Integral = this  { fix f :: "nat => real"    have "!! x. f sums x = (λ n. if even n then f (n div 2) else 0) sums x"    proof      fix x :: real assume "f sums x"      from sums_if[OF sums_zero this]      show "(λ n. if even n then f (n div 2) else 0) sums x" by auto    next      fix x :: real assume "(λ n. if even n then f (n div 2) else 0) sums x"      from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]      show "f sums x" unfolding sums_def by auto    qed    hence "op sums f = op sums (λ n. if even n then f (n div 2) else 0)" ..  } note sums_even = this  have Int_eq: "(∑ n. ?f n * real (Suc n) * x^n) = ?Int" unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "λ n. -1 ^ n * x ^ (2 * n)", symmetric]    by auto  { fix x :: real    have if_eq': "!! n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =      (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"      using n_even by auto    have idx_eq: "!! n. n * 2 + 1 = Suc (2 * n)" by auto    have "(∑ n. ?f n * x^(Suc n)) = ?arctan x" unfolding if_eq' idx_eq suminf_def sums_even[of "λ n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]      by auto  } note arctan_eq = this  have "DERIV (λ x. ∑ n. ?f n * x^(Suc n)) x :> (∑ n. ?f n * real (Suc n) * x^n)"  proof (rule DERIV_power_series')    show "x ∈ {- 1 <..< 1}" using `¦ x ¦ < 1` by auto    { fix x' :: real assume x'_bounds: "x' ∈ {- 1 <..< 1}"      hence "¦x'¦ < 1" by auto      let ?S = "∑ n. (-1)^n * x'^(2 * n)"      show "summable (λ n. ?f n * real (Suc n) * x'^n)" unfolding if_eq        by (rule sums_summable[where l="0 + ?S"], rule sums_if, rule sums_zero, rule summable_sums, rule summable_Integral[OF `¦x'¦ < 1`])    }  qed auto  thus ?thesis unfolding Int_eq arctan_eq .qedlemma arctan_series: assumes "¦ x ¦ ≤ 1"  shows "arctan x = (∑ k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "_ = suminf (λ n. ?c x n)")proof -  let "?c' x n" = "(-1)^n * x^(n*2)"  { fix r x :: real assume "0 < r" and "r < 1" and "¦ x ¦ < r"    have "¦x¦ < 1" using `r < 1` and `¦x¦ < r` by auto    from DERIV_arctan_series[OF this]    have "DERIV (λ x. suminf (?c x)) x :> (suminf (?c' x))" .  } note DERIV_arctan_suminf = this  { fix x :: real assume "¦x¦ ≤ 1" note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] }  note arctan_series_borders = this  { fix x :: real assume "¦x¦ < 1" have "arctan x = (∑ k. ?c x k)"  proof -    obtain r where "¦x¦ < r" and "r < 1" using dense[OF `¦x¦ < 1`] by blast    hence "0 < r" and "-r < x" and "x < r" by auto    have suminf_eq_arctan_bounded: "!! x a b. [| -r < a ; b < r ; a < b ; a ≤ x ; x ≤ b |] ==> suminf (?c x) - arctan x = suminf (?c a) - arctan a"    proof -      fix x a b assume "-r < a" and "b < r" and "a < b" and "a ≤ x" and "x ≤ b"      hence "¦x¦ < r" by auto      show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"      proof (rule DERIV_isconst2[of "a" "b"])        show "a < b" and "a ≤ x" and "x ≤ b" using `a < b` `a ≤ x` `x ≤ b` by auto        have "∀ x. -r < x ∧ x < r --> DERIV (λ x. suminf (?c x) - arctan x) x :> 0"        proof (rule allI, rule impI)          fix x assume "-r < x ∧ x < r" hence "¦x¦ < r" by auto          hence "¦x¦ < 1" using `r < 1` by auto          have "¦ - (x^2) ¦ < 1" using less_one_imp_sqr_less_one[OF `¦x¦ < 1`] by auto          hence "(λ n. (- (x^2)) ^ n) sums (1 / (1 - (- (x^2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)          hence "(?c' x) sums (1 / (1 - (- (x^2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto          hence suminf_c'_eq_geom: "inverse (1 + x^2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto          have "DERIV (λ x. suminf (?c x)) x :> (inverse (1 + x^2))" unfolding suminf_c'_eq_geom            by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `¦x¦ < r`])          from DERIV_add_minus[OF this DERIV_arctan]          show "DERIV (λ x. suminf (?c x) - arctan x) x :> 0" unfolding diff_minus by auto        qed        hence DERIV_in_rball: "∀ y. a ≤ y ∧ y ≤ b --> DERIV (λ x. suminf (?c x) - arctan x) y :> 0" using `-r < a` `b < r` by auto        thus "∀ y. a < y ∧ y < b --> DERIV (λ x. suminf (?c x) - arctan x) y :> 0" using `¦x¦ < r` by auto        show "∀ y. a ≤ y ∧ y ≤ b --> isCont (λ x. suminf (?c x) - arctan x) y" using DERIV_in_rball DERIV_isCont by auto      qed    qed    have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"      unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto    have "suminf (?c x) - arctan x = 0"    proof (cases "x = 0")      case True thus ?thesis using suminf_arctan_zero by auto    next      case False hence "0 < ¦x¦" and "- ¦x¦ < ¦x¦" by auto      have "suminf (?c (-¦x¦)) - arctan (-¦x¦) = suminf (?c 0) - arctan 0"        by (rule suminf_eq_arctan_bounded[where x="0" and a="-¦x¦" and b="¦x¦", symmetric])          (simp_all only: `¦x¦ < r` `-¦x¦ < ¦x¦` neg_less_iff_less)      moreover      have "suminf (?c x) - arctan x = suminf (?c (-¦x¦)) - arctan (-¦x¦)"        by (rule suminf_eq_arctan_bounded[where x="x" and a="-¦x¦" and b="¦x¦"])          (simp_all only: `¦x¦ < r` `-¦x¦ < ¦x¦` neg_less_iff_less)      ultimately      show ?thesis using suminf_arctan_zero by auto    qed    thus ?thesis by auto  qed } note when_less_one = this  show "arctan x = suminf (λ n. ?c x n)"  proof (cases "¦x¦ < 1")    case True thus ?thesis by (rule when_less_one)  next case False hence "¦x¦ = 1" using `¦x¦ ≤ 1` by auto    let "?a x n" = "¦1 / real (n*2+1) * x^(n*2+1)¦"    let "?diff x n" = "¦ arctan x - (∑ i = 0..<n. ?c x i)¦"    { fix n :: nat      have "0 < (1 :: real)" by auto      moreover      { fix x :: real assume "0 < x" and "x < 1" hence "¦x¦ ≤ 1" and "¦x¦ < 1" by auto        from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" by auto        note bounds = mp[OF arctan_series_borders(2)[OF `¦x¦ ≤ 1`] this, unfolded when_less_one[OF `¦x¦ < 1`, symmetric], THEN spec]        have "0 < 1 / real (n*2+1) * x^(n*2+1)" by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)        hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" by (rule abs_of_pos)        have "?diff x n ≤ ?a x n"        proof (cases "even n")          case True hence sgn_pos: "(-1)^n = (1::real)" by auto          from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto          from bounds[of m, unfolded this atLeastAtMost_iff]          have "¦arctan x - (∑i = 0..<n. (?c x i))¦ ≤ (∑i = 0..<n + 1. (?c x i)) - (∑i = 0..<n. (?c x i))" by auto          also have "… = ?c x n" unfolding One_nat_def by auto          also have "… = ?a x n" unfolding sgn_pos a_pos by auto          finally show ?thesis .        next          case False hence sgn_neg: "(-1)^n = (-1::real)" by auto          from `odd n` obtain m where m_def: "2 * m + 1 = n" unfolding odd_Suc_mult_two_ex by auto          hence m_plus: "2 * (m + 1) = n + 1" by auto          from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]          have "¦arctan x - (∑i = 0..<n. (?c x i))¦ ≤ (∑i = 0..<n. (?c x i)) - (∑i = 0..<n+1. (?c x i))" by auto          also have "… = - ?c x n" unfolding One_nat_def by auto          also have "… = ?a x n" unfolding sgn_neg a_pos by auto          finally show ?thesis .        qed        hence "0 ≤ ?a x n - ?diff x n" by auto      }      hence "∀ x ∈ { 0 <..< 1 }. 0 ≤ ?a x n - ?diff x n" by auto      moreover have "!!x. isCont (λ x. ?a x n - ?diff x n) x"        unfolding diff_minus divide_inverse        by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)      ultimately have "0 ≤ ?a 1 n - ?diff 1 n" by (rule LIM_less_bound)      hence "?diff 1 n ≤ ?a 1 n" by auto    }    have "?a 1 ----> 0"      unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def      by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)    have "?diff 1 ----> 0"    proof (rule LIMSEQ_I)      fix r :: real assume "0 < r"      obtain N :: nat where N_I: "!! n. N ≤ n ==> ?a 1 n < r" using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto      { fix n assume "N ≤ n" from `?diff 1 n ≤ ?a 1 n` N_I[OF this]        have "norm (?diff 1 n - 0) < r" by auto }      thus "∃ N. ∀ n ≥ N. norm (?diff 1 n - 0) < r" by blast    qed    from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]    have "(?c 1) sums (arctan 1)" unfolding sums_def by auto    hence "arctan 1 = (∑ i. ?c 1 i)" by (rule sums_unique)    show ?thesis    proof (cases "x = 1", simp add: `arctan 1 = (∑ i. ?c 1 i)`)      assume "x ≠ 1" hence "x = -1" using `¦x¦ = 1` by auto      have "- (pi / 2) < 0" using pi_gt_zero by auto      have "- (2 * pi) < 0" using pi_gt_zero by auto      have c_minus_minus: "!! i. ?c (- 1) i = - ?c 1 i" unfolding One_nat_def by auto      have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..      also have "… = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])      also have "… = - (arctan (tan (pi / 4)))" unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])      also have "… = - (arctan 1)" unfolding tan_45 ..      also have "… = - (∑ i. ?c 1 i)" using `arctan 1 = (∑ i. ?c 1 i)` by auto      also have "… = (∑ i. ?c (- 1) i)" using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]] unfolding c_minus_minus by auto      finally show ?thesis using `x = -1` by auto    qed  qedqedlemma arctan_half: fixes x :: real  shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x^2)))"proof -  obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" using tan_total by blast  hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto  have divide_nonzero_divide: "!! A B C :: real. C ≠ 0 ==> A / B = (A / C) / (B / C)" by auto  have "0 < cos y" using cos_gt_zero_pi[OF low high] .  hence "cos y ≠ 0" and cos_sqrt: "sqrt ((cos y) ^ 2) = cos y" by auto  have "1 + (tan y)^2 = 1 + sin y^2 / cos y^2" unfolding tan_def power_divide ..  also have "… = cos y^2 / cos y^2 + sin y^2 / cos y^2" using `cos y ≠ 0` by auto  also have "… = 1 / cos y^2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..  finally have "1 + (tan y)^2 = 1 / cos y^2" .  have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)" unfolding tan_def divide_nonzero_divide[OF `cos y ≠ 0`, symmetric] ..  also have "… = tan y / (1 + 1 / cos y)" using `cos y ≠ 0` unfolding add_divide_distrib by auto  also have "… = tan y / (1 + 1 / sqrt(cos y^2))" unfolding cos_sqrt ..  also have "… = tan y / (1 + sqrt(1 / cos y^2))" unfolding real_sqrt_divide by auto  finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)^2))" unfolding `1 + (tan y)^2 = 1 / cos y^2` .  have "arctan x = y" using arctan_tan low high y_eq by auto  also have "… = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto  also have "… = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half by auto  finally show ?thesis unfolding eq `tan y = x` .qedlemma arctan_monotone: assumes "x < y"  shows "arctan x < arctan y"  using assms by (simp only: arctan_less_iff)lemma arctan_monotone': assumes "x ≤ y" shows "arctan x ≤ arctan y"  using assms by (simp only: arctan_le_iff)lemma arctan_inverse:  assumes "x ≠ 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"proof (rule arctan_unique)  show "- (pi / 2) < sgn x * pi / 2 - arctan x"    using arctan_bounded [of x] assms    unfolding sgn_real_def    apply (auto simp add: algebra_simps)    apply (drule zero_less_arctan_iff [THEN iffD2])    apply arith    done  show "sgn x * pi / 2 - arctan x < pi / 2"    using arctan_bounded [of "- x"] assms    unfolding sgn_real_def arctan_minus    by auto  show "tan (sgn x * pi / 2 - arctan x) = 1 / x"    unfolding tan_inverse [of "arctan x", unfolded tan_arctan]    unfolding sgn_real_def    by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)qedtheorem pi_series: "pi / 4 = (∑ k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")proof -  have "pi / 4 = arctan 1" using arctan_one by auto  also have "… = ?SUM" using arctan_series[of 1] by auto  finally show ?thesis by autoqedsubsection {* 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: power_divide divide_le_eq not_sum_power2_lt_zero)donelemma 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)donelemma 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 (metis cos_minus minus_minus minus_mult_right sin_minus)donelemma 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)doneend`