(*  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 Transcendental

 imports Fact Series Deriv NthRoot

 begin

 subsection {* Properties of Power Series *}

 lemma lemma_realpow_diff:

   fixes y :: "'a::monoid_mult"

   shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"

 proof -

   assume "p \<le> n"

   hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)

   thus ?thesis by (simp add: power_commutes)

 qed

 lemma lemma_realpow_diff_sumr:

   fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows

      "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =

       y * (\<Sum>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) * (\<Sum>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: right_distrib del: setsum_op_ivl_Suc)

 apply (subst mult_left_commute [of "x - y"])

 apply (erule subst)

 apply (simp add: algebra_simps)

 done

 lemma lemma_realpow_rev_sumr:

      "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =

       (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"

 apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])

 apply (rule inj_onI, simp)

 apply auto

 apply (rule_tac x="n - x" in image_eqI, simp, simp)

 done

 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term

 x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}

 lemma powser_insidea:

   fixes x z :: "'a::{real_normed_field,banach}"

   assumes 1: "summable (\<lambda>n. f n * x ^ n)"

   assumes 2: "norm z < norm x"

   shows "summable (\<lambda>n. norm (f n * z ^ n))"

 proof -

   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp

   from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"

     by (rule summable_LIMSEQ_zero)

   hence "convergent (\<lambda>n. f n * x ^ n)"

     by (rule convergentI)

   hence "Cauchy (\<lambda>n. f n * x ^ n)"

     by (simp add: Cauchy_convergent_iff)

   hence "Bseq (\<lambda>n. f n * x ^ n)"

     by (rule Cauchy_Bseq)

   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"

     by (simp add: Bseq_def, safe)

   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>

                    K * norm (z ^ n) * inverse (norm (x ^ n))"

   proof (intro exI allI impI)

     fix n::nat assume "0 \<le> 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 "\<dots> \<le> K * norm (z ^ n)"

       by (simp only: mult_right_mono 4 norm_ge_zero)

     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"

       by (simp add: x_neq_0)

     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"

       by (simp only: mult_assoc)

     finally show "norm (norm (f n * z ^ n)) \<le>

                   K * norm (z ^ n) * inverse (norm (x ^ n))"

       by (simp add: mult_le_cancel_right x_neq_0)

   qed

   moreover have "summable (\<lambda>n. 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 (\<lambda>n. norm (z * inverse x) ^ n)"

       by (rule summable_geometric)

     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"

       by (rule summable_mult)

     thus "summable (\<lambda>n. 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 (\<lambda>n. norm (f n * z ^ n))"

     by (rule summable_comparison_test)

 qed

 lemma powser_inside:

   fixes f :: "nat \<Rightarrow> '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 \<Rightarrow> real" shows

   "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =

    (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"

 proof (induct n)

   case (Suc n)

   have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =

         (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> 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 "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto

   finally show ?case .

 qed auto

 lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"

   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"

   unfolding sums_def

 proof (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: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast

   let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"

   { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> 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 \<ge> 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 "\<dots> = ?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 "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast

 qed

 lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"

   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"

 proof -

   let ?s = "\<lambda> 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: "(\<lambda> 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: "\<And> n. Suc n - 1 = n" by auto

     have if_eq: "\<And>B T E. (if \<not> 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 "(\<lambda> 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 .

 qed

 subsection {* Alternating series test / Leibniz formula *}

 lemma sums_alternating_upper_lower:

   fixes a :: "nat \<Rightarrow> real"

   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"

   shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>

              ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"

   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")

 proof -

   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto

   have "\<forall> n. ?f n \<le> ?f (Suc n)"

   proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed

   moreover

   have "\<forall> n. ?g (Suc n) \<le> ?g n"

   proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]

     unfolding One_nat_def by auto qed

   moreover

   have "\<forall> n. ?f n \<le> ?g n"

   proof fix n show "?f n \<le> ?g n" using fg_diff a_pos

     unfolding One_nat_def by auto qed

   moreover

   have "(\<lambda> 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 "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto

     hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto

     thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto

   qed

   ultimately

   show ?thesis by (rule lemma_nest_unique)

 qed

 lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"

   assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"

   and a_monotone: "\<And> n. a (Suc n) \<le> a n"

   shows summable: "summable (\<lambda> n. (-1)^n * a n)"

   and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"

   and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"

   and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"

   and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"

 proof -

   let "?S n" = "(-1)^n * a n"

   let "?P n" = "\<Sum>i=0..<n. ?S i"

   let "?f n" = "?P (2 * n)"

   let "?g n" = "?P (2 * n + 1)"

   obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"

     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast

   let ?Sa = "\<lambda> m. \<Sum> 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: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto

     from `0 < r` `?g ----> l`[THEN LIMSEQ_D]

     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto

     { fix n :: nat

       assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 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 \<ge> 2 * f_no` have "n div 2 \<ge> 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 \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto

         from g[OF this]

         show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .

       qed

     }

     thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast

   qed

   hence sums_l: "(\<lambda>i. (-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 \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }

   { fix n show "?f n \<le> 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 auto

 qed

 theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"

   assumes a_zero: "a ----> 0" and "monoseq a"

   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")

   and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")

   and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")

   and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")

   and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")

 proof -

   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"

   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")

     case True

     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto

     { fix n have "a (Suc n) \<le> 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 \<le> a 0` by auto

   next

     let ?a = "\<lambda> n. - a n"

     case False

     with monoseq_le[OF `monoseq a` `a ----> 0`]

     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto

     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto

     { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }

     note monotone = this

     note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF LIMSEQ_minus[OF `a ----> 0`, unfolded minus_zero] monotone]

     have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto

     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto

     from this[THEN sums_minus]

     have "(\<lambda> n. (-1)^n * a n) sums -l" by auto

     hence ?summable unfolding summable_def by auto

     moreover

     have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto

     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]

     have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto

     have ?pos using `0 \<le> ?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 LIMSEQ_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 .

 qed

 subsection {* Term-by-Term Differentiability of Power Series *}

 definition

   diffs :: "(nat => 'a::ring_1) => nat => 'a" where

   "diffs c = (%n. of_nat (Suc n) * c(Suc n))"

 text{*Lemma about distributing negation over it*}

 lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"

 by (simp add: diffs_def)

 lemma sums_Suc_imp:

   assumes f: "f 0 = 0"

   shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"

 unfolding sums_def

 apply (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)

 done

 lemma 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

          (\<Sum>n. (diffs c)(n) * (x ^ n))"

 unfolding diffs_def

 apply (drule summable_sums)

 apply (rule sums_Suc_imp, simp_all)

 done

 lemma lemma_termdiff1:

   fixes z :: "'a :: {monoid_mult,comm_ring}" shows

   "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =

    (\<Sum>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) = (\<Sum>i = 0..<n. f i - r)"

 by (simp add: setsum_subtractf)

 lemma lemma_termdiff2:

   fixes h :: "'a :: {field}"

   assumes h: "h \<noteq> 0" shows

   "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =

    h * (\<Sum>p=0..< n - Suc 0. \<Sum>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 simp

 apply (simp only: lemma_termdiff1 setsum_right_distrib)

 apply (rule setsum_cong [OF refl])

 apply (simp add: diff_minus [symmetric] less_iff_Suc_add)

 apply (clarify)

 apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac

             del: setsum_op_ivl_Suc power_Suc)

 apply (subst mult_assoc [symmetric], subst power_add [symmetric])

 apply (simp add: mult_ac)

 done

 lemma real_setsum_nat_ivl_bounded2:

   fixes K :: "'a::linordered_semidom"

   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"

   assumes K: "0 \<le> K"

   shows "setsum f {0..<n-k} \<le> of_nat n * K"

 apply (rule order_trans [OF setsum_mono])

 apply (rule f, simp)

 apply (simp add: mult_right_mono K)

 done

 lemma lemma_termdiff3:

   fixes h z :: "'a::{real_normed_field}"

   assumes 1: "h \<noteq> 0"

   assumes 2: "norm z \<le> K"

   assumes 3: "norm (z + h) \<le> K"

   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))

           \<le> 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 (\<Sum>p = 0..<n - Suc 0. \<Sum>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 "\<dots> \<le> 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 \<le> K" by (rule order_trans)

     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> 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 (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.

               (z + h) ^ q * z ^ (n - 2 - q))

           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"

       apply (intro

          order_trans [OF norm_setsum]

          real_setsum_nat_ivl_bounded2

          mult_nonneg_nonneg

          zero_le_imp_of_nat

          zero_le_power K)

       apply (rule le_Kn, simp)

       done

   qed

   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"

     by (simp only: mult_assoc)

   finally show ?thesis .

 qed

 lemma lemma_termdiff4:

   fixes f :: "'a::{real_normed_field} \<Rightarrow>

               'b::real_normed_vector"

   assumes k: "0 < (k::real)"

   assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"

   shows "f -- 0 --> 0"

 unfolding LIM_eq diff_0_right

 proof (safe)

   let ?h = "of_real (k / 2)::'a"

   have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all

   hence "norm (f ?h) \<le> K * norm ?h" by (rule le)

   hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])

   hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)

   fix r::real assume r: "0 < r"

   show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"

   proof (cases)

     assume "K = 0"

     with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"

       by simp

     thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..

   next

     assume K_neq_zero: "K \<noteq> 0"

     with zero_le_K have K: "0 < K" by simp

     show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> 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 \<noteq> 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) \<le> 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 "\<dots> = r / 2"

         using K_neq_zero by simp

       also have "r / 2 < r"

         using r by simp

       finally show "norm (f x) < r" .

     qed

   qed

 qed

 lemma lemma_termdiff5:

   fixes g :: "'a::{real_normed_field} \<Rightarrow>

               nat \<Rightarrow> 'b::banach"

   assumes k: "0 < (k::real)"

   assumes f: "summable f"

   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"

   shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"

 proof (rule lemma_termdiff4 [OF k])

   fix h::'a assume "h \<noteq> 0" and "norm h < k"

   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"

     by (simp add: le)

   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"

     by simp

   moreover from f have B: "summable (\<lambda>n. f n * norm h)"

     by (rule summable_mult2)

   ultimately have C: "summable (\<lambda>n. norm (g h n))"

     by (rule summable_comparison_test)

   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"

     by (rule summable_norm)

   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"

     by (rule summable_le)

   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"

     by (rule suminf_mult2 [symmetric])

   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .

 qed

 text{* FIXME: Long proofs*}

 lemma termdiffs_aux:

   fixes x :: "'a::{real_normed_field,banach}"

   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"

   assumes 2: "norm x < norm K"

   shows "(\<lambda>h. \<Sum>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 \<noteq> 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 (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"

       by (rule powser_insidea)

     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"

       using r

       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)

     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"

       by (rule diffs_equiv [THEN sums_summable])

     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))

       = (\<lambda>n. 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

       (\<lambda>n. 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

       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *

            r ^ (n - Suc 0)) =

        (\<lambda>n. 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 (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .

   next

     fix h::'a and n::nat

     assume h: "h \<noteq> 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)))

           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"

       apply (simp only: norm_mult mult_assoc)

       apply (rule mult_left_mono [OF _ norm_ge_zero])

       apply (simp (no_asm) add: mult_assoc [symmetric])

       apply (rule lemma_termdiff3)

       apply (rule h)

       apply (rule r1 [THEN order_less_imp_le])

       apply (rule xh [THEN order_less_imp_le])

       done

   qed

 qed

 lemma termdiffs:

   fixes K x :: "'a::{real_normed_field,banach}"

   assumes 1: "summable (\<lambda>n. c n * K ^ n)"

   assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"

   assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"

   assumes 4: "norm x < norm K"

   shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"

 unfolding deriv_def

 proof (rule LIM_zero_cancel)

   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h

             - suminf (\<lambda>n. 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 \<noteq> 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 (\<lambda>n. c n * x ^ n)"

       by (rule powser_inside [OF 1 4])

     have B: "summable (\<lambda>n. c n * (x + h) ^ n)"

       by (rule powser_inside [OF 1 5])

     have C: "summable (\<lambda>n. diffs c n * x ^ n)"

       by (rule powser_inside [OF 2 4])

     show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h

              - (\<Sum>n. diffs c n * x ^ n) =

           (\<Sum>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 "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -

                of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"

         by (rule termdiffs_aux [OF 3 4])

   qed

 qed

 subsection {* Derivability of power series *}

 lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"

   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"

   and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"

   and "summable (f' x0)"

   and "summable L" and L_def: "\<And> n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar> f x n - f y n \<bar> \<le> L n * \<bar> x - y \<bar>"

   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"

   unfolding deriv_def

 proof (rule LIM_I)

   fix r :: real assume "0 < r" hence "0 < r/3" by auto

   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"

     using suminf_exist_split[OF `0 < r/3` `summable L`] by auto

   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"

     using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto

   let ?N = "Suc (max N_L N_f')"

   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and

     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < 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 \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"

   def S' \<equiv> "Min (?s ` { 0 ..< ?N })"

   have "0 < S'" unfolding S'_def

   proof (rule iffD2[OF Min_gr_iff])

     show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"

     proof (rule ballI)

       fix x assume "x \<in> ?s ` {0..<?N}"

       then obtain n where "x = ?s n" and "n \<in> {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 \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?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 \<equiv> "min (min (x0 - a) (b - x0)) S'"

   hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" and "S \<le> S'" using x0_in_I and `0 < S'`

     by auto

   { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"

     hence x_in_I: "x0 + x \<in> { 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 "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"

         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .

       hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 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 "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .

     hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto

     have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..

     also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"

     proof (rule setsum_strict_mono)

       fix n assume "n \<in> { 0 ..< ?N}"

       have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .

       also have "S \<le> S'" using `S \<le> S'` .

       also have "S' \<le> ?s n" unfolding S'_def

       proof (rule Min_le_iff[THEN iffD2])

         have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto

         thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast

       qed auto

       finally have "\<bar> x \<bar> < ?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 "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .

       with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`

       show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast

     qed auto

     also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)

     also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto

     also have "\<dots> = r/3" by auto

     finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .

     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]

     have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =

                     \<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto

     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" 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 "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto

     also have "\<dots> < r /3 + r/3 + r/3"

       using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`

       by (rule add_strict_mono [OF add_less_le_mono])

     finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"

       by auto

   } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>

       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`

     unfolding real_norm_def diff_0_right by blast

 qed

 lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"

   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"

   and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"

   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"

   (is "DERIV (\<lambda> 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 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto

     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"

     proof (rule DERIV_series')

       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"

       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 \<in> {-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 \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"

         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"

         proof -

           have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"

             unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto

           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"

           proof (rule mult_left_mono)

             have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)

             also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"

             proof (rule setsum_mono)

               fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto

               { fix n fix x :: real assume "x \<in> {-R'<..<R'}"

                 hence "\<bar>x\<bar> \<le> R'"  by auto

                 hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)

               } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`

               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto

               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto

             qed

             also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto

             finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .

             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto

           qed

           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra

           finally show ?thesis .

         qed }

       { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"

           by (auto intro!: DERIV_intros simp del: power_Suc) }

       { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto

         have "summable (\<lambda> n. f n * x^n)"

         proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)

           fix n

           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)

           show "\<bar>f n * x ^ n\<bar> \<le> 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 \<in> {-R <..< R}`] .

       show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .

     qed

   } note for_subinterval = this

   let ?R = "(R + \<bar>x0\<bar>) / 2"

   have "\<bar>x0\<bar> < ?R" using assms by auto

   hence "- ?R < x0"

   proof (cases "x0 < 0")

     case True

     hence "- x0 < ?R" using `\<bar>x0\<bar> < ?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 "\<dots> \<le> 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 .

 qed

 subsection {* Exponential Function *}

 definition

   exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where

   "exp x = (\<Sum>n. x ^ n /\<^sub>R real (fact n))"

 lemma summable_exp_generic:

   fixes x :: "'a::{real_normed_algebra_1,banach}"

   defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"

   shows "summable S"

 proof -

   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>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 \<le> n"

     have "norm x \<le> real N * r"

       using N by (rule order_less_imp_le)

     also have "real N * r \<le> real (Suc n) * r"

       using r0 n by (simp add: mult_right_mono)

     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"

       using norm_ge_zero by (rule mult_right_mono)

     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"

       by (rule order_trans [OF norm_mult_ineq])

     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"

       by (simp add: pos_divide_le_eq mult_ac)

     thus "norm (S (Suc n)) \<le> r * norm (S n)"

       by (simp add: S_Suc inverse_eq_divide)

   qed

 qed

 lemma summable_norm_exp:

   fixes x :: "'a::{real_normed_algebra_1,banach}"

   shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"

 proof (rule summable_norm_comparison_test [OF exI, rule_format])

   show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"

     by (rule summable_exp_generic)

 next

   fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"

     by (simp add: norm_power_ineq)

 qed

 lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"

 by (insert summable_exp_generic [where x=x], simp)

 lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>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 (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"

 by (simp add: diffs_def)

 lemma lemma_exp_ext: "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"

 by (auto simp add: exp_def)

 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"

 apply (simp add: exp_def)

 apply (subst lemma_exp_ext)

 apply (subgoal_tac "DERIV (\<lambda>u. \<Sum>n. of_real (inverse (real (fact n))) * u ^ n) x :> (\<Sum>n. diffs (\<lambda>n. of_real (inverse (real (fact n)))) n * x ^ n)")

 apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs)

 apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)

 apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+

 apply (simp del: of_real_add)

 done

 lemma isCont_exp [simp]: "isCont exp x"

 by (rule DERIV_exp [THEN DERIV_isCont])

 subsubsection {* Properties of the Exponential Function *}

 lemma powser_zero:

   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"

   shows "(\<Sum>n. f n * 0 ^ n) = f 0"

 proof -

   have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"

     by (rule sums_unique [OF series_zero], simp add: power_0_left)

   thus ?thesis unfolding One_nat_def by simp

 qed

 lemma exp_zero [simp]: "exp 0 = 1"

 unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)

 lemma setsum_cl_ivl_Suc2:

   "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>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 \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"

   shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"

 proof (induct n)

   case 0

   show ?case

     unfolding S_def by simp

 next

   case (Suc n)

   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"

     unfolding S_def by (simp del: mult_Suc)

   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"

     by simp

   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"

     by (simp only: times_S)

   also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"

     by (simp only: Suc)

   also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))

                 + y * (\<Sum>i=0..n. S x i * S y (n-i))"

     by (rule left_distrib)

   also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))

                 + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"

     by (simp only: setsum_right_distrib mult_ac)

   also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))

                 + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"

     by (simp add: times_S Suc_diff_le)

   also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =

              (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"

     by (subst setsum_cl_ivl_Suc2, simp)

   also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =

              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"

     by (subst setsum_cl_ivl_Suc, simp)

   also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +

              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =

              (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>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 "\<dots> = real (Suc n) *\<^sub>R (\<Sum>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) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"

     by (simp del: setsum_cl_ivl_Suc)

 qed

 lemma exp_add: "exp (x + y) = exp x * exp y"

 unfolding exp_def

 by (simp only: Cauchy_product summable_norm_exp exp_series_add)

 lemma 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_def

 apply (subst suminf_of_real)

 apply (rule summable_exp_generic)

 apply (simp add: scaleR_conv_of_real)

 done

 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"

 proof

   have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)

   also assume "exp x = 0"

   finally show "False" by simp

 qed

 lemma 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 \<le> exp (x::real)"

 proof -

   have "0 \<le> exp (x/2) * exp (x/2)" by simp

   thus ?thesis by (simp add: exp_add [symmetric])

 qed

 lemma exp_gt_zero [simp]: "0 < exp (x::real)"

 by (simp add: order_less_le)

 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"

 by (simp add: not_less)

 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"

 by (simp add: not_le)

 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"

 by simp

 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"

 apply (induct "n")

 apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)

 done

 text {* Strict monotonicity of exponential. *}

 lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> 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)

 done

 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"

 proof -

   assume x: "0 < x"

   hence "1 < 1 + x" by simp

   also from x have "1 + x \<le> exp x"

     by (simp add: exp_ge_add_one_self_aux)

   finally show ?thesis .

 qed

 lemma 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 simp

 qed

 lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"

 apply (simp add: linorder_not_le [symmetric])

 apply (auto simp add: order_le_less exp_less_mono)

 done

 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"

 by (auto intro: exp_less_mono exp_less_cancel)

 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"

 by (auto simp add: linorder_not_less [symmetric])

 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> 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) \<longleftrightarrow> 0 < x"

   using exp_less_cancel_iff [where x=0 and y=x] by simp

 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"

   using exp_less_cancel_iff [where x=x and y=0] by simp

 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"

   using exp_le_cancel_iff [where x=0 and y=x] by simp

 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"

   using exp_le_cancel_iff [where x=x and y=0] by simp

 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"

   using exp_inj_iff [where x=x and y=0] by simp

 lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"

 apply (rule IVT)

 apply (auto intro: isCont_exp simp add: le_diff_eq)

 apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")

 apply simp

 apply (rule exp_ge_add_one_self_aux, simp)

 done

 lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"

 apply (rule_tac x = 1 and y = y in linorder_cases)

 apply (drule order_less_imp_le [THEN lemma_exp_total])

 apply (rule_tac [2] x = 0 in exI)

 apply (frule_tac [3] one_less_inverse)

 apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)

 apply (rule_tac x = "-x" in exI)

 apply (simp add: exp_minus)

 done

 subsection {* 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 \<Longrightarrow> exp (ln x) = x"

 by (auto dest: exp_total)

 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"

 apply (rule iffI)

 apply (erule subst, rule exp_gt_zero)

 apply (erule exp_ln)

 done

 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"

 by (erule subst, rule ln_exp)

 lemma ln_one [simp]: "ln 1 = 0"

 by (rule ln_unique, simp)

 lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"

 by (rule ln_unique, simp add: exp_add)

 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"

 by (rule ln_unique, simp add: exp_minus)

 lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"

 by (rule ln_unique, simp add: exp_diff)

 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"

 by (rule ln_unique, simp add: exp_real_of_nat_mult)

 lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"

 by (subst exp_less_cancel_iff [symmetric], simp)

 lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"

 by (simp add: linorder_not_less [symmetric])

 lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"

 by (simp add: order_eq_iff)

 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"

 apply (rule exp_le_cancel_iff [THEN iffD1])

 apply (simp add: exp_ge_add_one_self_aux)

 done

 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"

 by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all

 lemma ln_ge_zero [simp]:

   assumes x: "1 \<le> x" shows "0 \<le> ln x"

 proof -

   have "0 < x" using x by arith

   hence "exp 0 \<le> exp (ln x)"

     by (simp add: x)

   thus ?thesis by (simp only: exp_le_cancel_iff)

 qed

 lemma ln_ge_zero_imp_ge_one:

   assumes ln: "0 \<le> ln x"

       and x:  "0 < x"

   shows "1 \<le> x"

 proof -

   from ln have "ln 1 \<le> ln x" by simp

   thus ?thesis by (simp add: x del: ln_one)

 qed

 lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"

 by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)

 lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"

 by (insert ln_ge_zero_iff [of x], arith)

 lemma ln_gt_zero:

   assumes x: "1 < x" shows "0 < ln x"

 proof -

   have "0 < x" using x by arith

   hence "exp 0 < exp (ln x)" by (simp add: x)

   thus ?thesis  by (simp only: exp_less_cancel_iff)

 qed

 lemma ln_gt_zero_imp_gt_one:

   assumes ln: "0 < ln x"

       and x:  "0 < x"

   shows "1 < x"

 proof -

   from ln have "ln 1 < ln x" by simp

   thus ?thesis by (simp add: x del: ln_one)

 qed

 lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"

 by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)

 lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"

 by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)

 lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"

 by simp

 lemma exp_ln_eq: "exp u = x ==> ln x = u"

 by auto

 lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"

 apply (subgoal_tac "isCont ln (exp (ln x))", simp)

 apply (rule isCont_inverse_function [where f=exp], simp_all)

 done

 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"

 apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])

 apply (erule lemma_DERIV_subst [OF DERIV_exp exp_ln])

 apply (simp_all add: abs_if isCont_ln)

 done

 lemma 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 = (\<Sum> 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 \<in> {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 "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)

     also have "\<dots> = 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: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto

     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"

     proof (rule DERIV_power_series')

       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto

       { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto

         show "summable (\<lambda>n. -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 (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto

     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .

     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"

       by (rule DERIV_diff)

     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto

   qed (auto simp add: assms)

   thus ?thesis by auto

 qed

 subsection {* Sine and Cosine *}

 definition

   sin_coeff :: "nat \<Rightarrow> real" where

   "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"

 definition

   cos_coeff :: "nat \<Rightarrow> real" where

   "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"

 definition

   sin :: "real => real" where

   "sin x = (\<Sum>n. sin_coeff n * x ^ n)"

 definition

   cos :: "real => real" where

   "cos x = (\<Sum>n. cos_coeff n * x ^ n)"

 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"

 unfolding sin_coeff_def

 apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)

 apply (rule_tac [2] summable_exp)

 apply (rule_tac x = 0 in exI)

 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)

 done

 lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"

 unfolding cos_coeff_def

 apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)

 apply (rule_tac [2] summable_exp)

 apply (rule_tac x = 0 in exI)

 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)

 done

 lemma lemma_STAR_sin:

      "(if even n then 0

        else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"

 by (induct "n", auto)

 lemma lemma_STAR_cos:

      "0 < n -->

       -1 ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"

 by (induct "n", auto)

 lemma lemma_STAR_cos1:

      "0 < n -->

       (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"

 by (induct "n", auto)

 lemma lemma_STAR_cos2:

   "(\<Sum>n=1..<n. if even n then -1 ^ (n div 2)/(real (fact n)) *  0 ^ n

                          else 0) = 0"

 apply (induct "n")

 apply (case_tac [2] "n", auto)

 done

 lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"

 unfolding sin_def by (rule summable_sin [THEN summable_sums])

 lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"

 unfolding cos_def by (rule summable_cos [THEN summable_sums])

 lemma sin_fdiffs: "diffs sin_coeff = cos_coeff"

 unfolding sin_coeff_def cos_coeff_def

 by (auto intro!: ext

          simp add: diffs_def divide_inverse real_of_nat_def of_nat_mult

          simp del: mult_Suc of_nat_Suc)

 lemma sin_fdiffs2: "diffs sin_coeff n = cos_coeff n"

 by (simp only: sin_fdiffs)

 lemma cos_fdiffs: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"

 unfolding sin_coeff_def cos_coeff_def

 by (auto intro!: ext

          simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def of_nat_mult

          simp del: mult_Suc of_nat_Suc)

 lemma cos_fdiffs2: "diffs cos_coeff n = - sin_coeff n"

 by (simp only: cos_fdiffs)

 text{*Now at last we can get the derivatives of exp, sin and cos*}

 lemma lemma_sin_minus: "- sin x = (\<Sum>n. - (sin_coeff n * x ^ n))"

 by (auto intro!: sums_unique sums_minus sin_converges)

 lemma lemma_sin_ext: "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"

   by (auto simp add: sin_def)

 lemma lemma_cos_ext: "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"

   by (auto simp add: cos_def)

 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"

 apply (simp add: cos_def)

 apply (subst lemma_sin_ext)

 apply (auto simp add: sin_fdiffs2 [symmetric])

 apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)

 apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)

 done

 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"

 apply (subst lemma_cos_ext)

 apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)

 apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)

 apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus)

 done

 lemma isCont_sin [simp]: "isCont sin x"

 by (rule DERIV_sin [THEN DERIV_isCont])

 lemma isCont_cos [simp]: "isCont cos x"

 by (rule DERIV_cos [THEN DERIV_isCont])

 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 DERIV_sin_sin_mult [simp]:

      "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"

 by (rule DERIV_mult, auto)

 lemma DERIV_sin_sin_mult2 [simp]:

      "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"

 apply (cut_tac x = x in DERIV_sin_sin_mult)

 apply (auto simp add: mult_assoc)

 done

 lemma DERIV_sin_realpow2 [simp]:

      "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"

 by (auto simp add: numeral_2_eq_2 mult_assoc [symmetric])

 lemma DERIV_sin_realpow2a [simp]:

      "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"

 by (auto simp add: numeral_2_eq_2)

 lemma DERIV_cos_cos_mult [simp]:

      "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"

 by (rule DERIV_mult, auto)

 lemma DERIV_cos_cos_mult2 [simp]:

      "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"

 apply (cut_tac x = x in DERIV_cos_cos_mult)

 apply (auto simp add: mult_ac)

 done

 lemma DERIV_cos_realpow2 [simp]:

      "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"

 by (auto simp add: numeral_2_eq_2 mult_assoc [symmetric])

 lemma DERIV_cos_realpow2a [simp]:

      "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"

 by (auto simp add: numeral_2_eq_2)

 lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"

 by auto

 lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"

   by (auto intro!: DERIV_intros)

 (* most useful *)

 lemma DERIV_cos_cos_mult3 [simp]:

      "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"

   by (auto intro!: DERIV_intros)

 lemma DERIV_sin_circle_all:

      "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>

              (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"

   by (auto intro!: DERIV_intros)

 lemma DERIV_sin_circle_all_zero [simp]:

      "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"

 by (cut_tac DERIV_sin_circle_all, auto)

 lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"

 apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])

 apply (auto simp add: numeral_2_eq_2)

 done

 lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"

 apply (subst add_commute)

 apply (rule sin_cos_squared_add)

 done

 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"

 apply (cut_tac x = x in sin_cos_squared_add2)

 apply (simp add: power2_eq_square)

 done

 lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"

 apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])

 apply simp

 done

 lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"

 apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])

 apply simp

 done

 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"

 by (rule power2_le_imp_le, simp_all add: sin_squared_eq)

 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"

 apply (insert abs_sin_le_one [of x])

 apply (simp add: abs_le_iff del: abs_sin_le_one)

 done

 lemma sin_le_one [simp]: "sin x \<le> 1"

 apply (insert abs_sin_le_one [of x])

 apply (simp add: abs_le_iff del: abs_sin_le_one)

 done

 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"

 by (rule power2_le_imp_le, simp_all add: cos_squared_eq)

 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"

 apply (insert abs_cos_le_one [of x])

 apply (simp add: abs_le_iff del: abs_cos_le_one)

 done

 lemma cos_le_one [simp]: "cos x \<le> 1"

 apply (insert abs_cos_le_one [of x])

 apply (simp add: abs_le_iff del: abs_cos_le_one)

 done

 lemma DERIV_fun_pow: "DERIV g x :> m ==>

       DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"

 unfolding One_nat_def

 apply (rule lemma_DERIV_subst)

 apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)

 apply (rule DERIV_pow, auto)

 done

 lemma DERIV_fun_exp:

      "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"

 apply (rule lemma_DERIV_subst)

 apply (rule_tac f = exp in DERIV_chain2)

 apply (rule DERIV_exp, auto)

 done

 lemma DERIV_fun_sin:

      "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"

 apply (rule lemma_DERIV_subst)

 apply (rule_tac f = sin in DERIV_chain2)

 apply (rule DERIV_sin, auto)

 done

 lemma DERIV_fun_cos:

      "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"

 apply (rule lemma_DERIV_subst)

 apply (rule_tac f = cos in DERIV_chain2)

 apply (rule DERIV_cos, auto)

 done

 (* lemma *)

 lemma lemma_DERIV_sin_cos_add:

      "\<forall>x.

          DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +

                (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"

   by (auto intro!: DERIV_intros simp add: algebra_simps)

 lemma sin_cos_add [simp]:

      "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +

       (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"

 apply (cut_tac y = 0 and x = x and y7 = y

        in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])

 apply (auto simp add: numeral_2_eq_2)

 done

 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"

 apply (cut_tac x = x and y = y in sin_cos_add)

 apply (simp del: sin_cos_add)

 done

 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"

 apply (cut_tac x = x and y = y in sin_cos_add)

 apply (simp del: sin_cos_add)

 done

 lemma lemma_DERIV_sin_cos_minus:

     "\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"

   by (auto intro!: DERIV_intros simp add: algebra_simps)

 lemma sin_cos_minus:

     "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"

 apply (cut_tac y = 0 and x = x

        in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])

 apply simp

 done

 lemma sin_minus [simp]: "sin (-x) = -sin(x)"

   using sin_cos_minus [where x=x] by simp

 lemma cos_minus [simp]: "cos (-x) = cos(x)"

   using sin_cos_minus [where x=x] by simp

 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"

 by (simp add: diff_minus sin_add)

 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"

 by (simp add: sin_diff mult_commute)

 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"

 by (simp add: diff_minus cos_add)

 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"

 by (simp add: cos_diff mult_commute)

 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"

   using sin_add [where x=x and y=x] by simp

 lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"

   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 \<le> (x::real) & x \<le> 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 "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"

     unfolding sin_def

     by (rule sin_converges [THEN sums_summable, THEN sums_group], simp)

   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)

 qed

 text {* FIXME: This is a long, ugly proof! *}

 lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"

 apply (subgoal_tac

        "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.

               -1 ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1))

      sums (\<Sum>n. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")

  prefer 2

  apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp)

 apply (rotate_tac 2)

 apply (drule sin_paired [THEN sums_unique, THEN ssubst])

 unfolding One_nat_def

 apply (auto simp del: fact_Suc)

 apply (frule sums_unique)

 apply (auto simp del: fact_Suc)

 apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])

 apply (auto simp del: fact_Suc)

 apply (erule sums_summable)

 apply (case_tac "m=0")

 apply (simp (no_asm_simp))

 apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")

 apply (simp only: mult_less_cancel_left, simp)

 apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])

 apply (subgoal_tac "x*x < 2*3", simp)

 apply (rule mult_strict_mono)

 apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)

 apply (subst fact_Suc)

 apply (subst fact_Suc)

 apply (subst fact_Suc)

 apply (subst fact_Suc)

 apply (subst real_of_nat_mult)

 apply (subst real_of_nat_mult)

 apply (subst real_of_nat_mult)

 apply (subst real_of_nat_mult)

 apply (simp (no_asm) add: divide_inverse del: fact_Suc)

 apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)

 apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right)

 apply (auto simp add: mult_assoc simp del: fact_Suc)

 apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right)

 apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)

 apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)")

 apply (erule ssubst)+

 apply (auto simp del: fact_Suc)

 apply (subgoal_tac "0 < x ^ (4 * m) ")

  prefer 2 apply (simp only: zero_less_power)

 apply (simp (no_asm_simp) add: mult_less_cancel_left)

 apply (rule mult_strict_mono)

 apply (simp_all (no_asm_simp))

 done

 lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"

 by (auto intro: sin_gt_zero)

 lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"

 apply (cut_tac x = x in sin_gt_zero1)

 apply (auto simp add: cos_squared_eq cos_double)

 done

 lemma cos_paired:

      "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"

 proof -

   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"

     unfolding cos_def

     by (rule cos_converges [THEN sums_summable, THEN sums_group], simp)

   thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)

 qed

 lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"

 by simp

 lemma 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)

 done

 lemma 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)

 done

 lemma 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 =

  "\<Sum>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_def

 apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]

             del: fact_Suc)

 apply (rule real_mult_inverse_cancel2)

 apply (simp del: fact_Suc)

 apply (simp del: fact_Suc)

 apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)

 apply (subst fact_lemma)

 apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])

 apply (simp only: real_of_nat_mult)

 apply (rule mult_strict_mono, force)

   apply (rule_tac [3] real_of_nat_ge_zero)

  prefer 2 apply force

 apply (rule real_of_nat_less_iff [THEN iffD2])

 apply (rule fact_less_mono_nat, auto)

 done

 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]

 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]

 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"

 apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")

 apply (rule_tac [2] IVT2)

 apply (auto intro: DERIV_isCont DERIV_cos)

 apply (cut_tac x = xa and y = y in linorder_less_linear)

 apply (rule ccontr)

 apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")

 apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)

 apply (drule_tac f = cos in Rolle)

 apply (drule_tac [5] f = cos in Rolle)

 apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)

 apply (metis order_less_le_trans less_le sin_gt_zero)

 apply (metis order_less_le_trans less_le sin_gt_zero)

 done

 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"

 by (simp add: pi_def)

 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"

 by (simp add: pi_half cos_is_zero [THEN theI'])

 lemma pi_half_gt_zero [simp]: "0 < pi / 2"

 apply (rule order_le_neq_trans)

 apply (simp add: pi_half cos_is_zero [THEN theI'])

 apply (rule notI, drule arg_cong [where f=cos], simp)

 done

 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]

 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]

 lemma pi_half_less_two [simp]: "pi / 2 < 2"

 apply (rule order_le_neq_trans)

 apply (simp add: pi_half cos_is_zero [THEN theI'])

 apply (rule notI, drule arg_cong [where f=cos], simp)

 done

 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]

 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]

 lemma pi_gt_zero [simp]: "0 < pi"

 by (insert pi_half_gt_zero, simp)

 lemma pi_ge_zero [simp]: "0 \<le> pi"

 by (rule pi_gt_zero [THEN order_less_imp_le])

 lemma pi_neq_zero [simp]: "pi \<noteq> 0"

 by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])

 lemma pi_not_less_zero [simp]: "\<not> pi < 0"

 by (simp add: linorder_not_less)

 lemma minus_pi_half_less_zero: "-(pi/2) < 0"

 by simp

 lemma m2pi_less_pi: "- (2 * pi) < pi"

 proof -

   have "- (2 * pi) < 0" and "0 < pi" by auto

   from order_less_trans[OF this] show ?thesis .

 qed

 lemma sin_pi_half [simp]: "sin(pi/2) = 1"

 apply (cut_tac x = "pi/2" in sin_cos_squared_add2)

 apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])

 apply (simp add: power2_eq_1_iff)

 done

 lemma cos_pi [simp]: "cos pi = -1"

 by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)

 lemma sin_pi [simp]: "sin pi = 0"

 by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)

 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"

 by (simp add: diff_minus cos_add)

 declare sin_cos_eq [symmetric, simp]

 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"

 by (simp add: cos_add)

 declare minus_sin_cos_eq [symmetric, simp]

 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"

 by (simp add: diff_minus sin_add)

 declare cos_sin_eq [symmetric, simp]

 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"

 by (simp add: sin_add)

 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"

 by (simp add: sin_add)

 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"

 by (simp add: cos_add)

 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"

 by (simp add: sin_add cos_double)

 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"

 by (simp add: cos_add cos_double)

 lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"

 apply (induct "n")

 apply (auto simp add: real_of_nat_Suc left_distrib)

 done

 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"

 proof -

   have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)

   also have "... = -1 ^ n" by (rule cos_npi)

   finally show ?thesis .

 qed

 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"

 apply (induct "n")

 apply (auto simp add: real_of_nat_Suc left_distrib)

 done

 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"

 by (simp add: mult_commute [of pi])

 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"

 by (simp add: cos_double)

 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"

 by simp

 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"

 apply (rule sin_gt_zero, assumption)

 apply (rule order_less_trans, assumption)

 apply (rule pi_half_less_two)

 done

 lemma sin_less_zero:

   assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"

 proof -

   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)

   thus ?thesis by simp

 qed

 lemma pi_less_4: "pi < 4"

 by (cut_tac pi_half_less_two, auto)

 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"

 apply (cut_tac pi_less_4)

 apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)

 apply (cut_tac cos_is_zero, safe)

 apply (rename_tac y z)

 apply (drule_tac x = y in spec)

 apply (drule_tac x = "pi/2" in spec, simp)

 done

 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"

 apply (rule_tac x = x and y = 0 in linorder_cases)

 apply (rule cos_minus [THEN subst])

 apply (rule cos_gt_zero)

 apply (auto intro: cos_gt_zero)

 done

 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"

 apply (auto simp add: order_le_less cos_gt_zero_pi)

 apply (subgoal_tac "x = pi/2", auto)

 done

 lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"

 apply (subst sin_cos_eq)

 apply (rotate_tac 1)

 apply (drule real_sum_of_halves [THEN ssubst])

 apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])

 done

 lemma pi_ge_two: "2 \<le> pi"

 proof (rule ccontr)

   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto

   have "\<exists>y > pi. y < 2 \<and> 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 auto

 qed

 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"

 by (auto simp add: order_le_less sin_gt_zero_pi)

 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"

 apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")

 apply (rule_tac [2] IVT2)

 apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)

 apply (cut_tac x = xa and y = y in linorder_less_linear)

 apply (rule ccontr, auto)

 apply (drule_tac f = cos in Rolle)

 apply (drule_tac [5] f = cos in Rolle)

 apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos

             dest!: DERIV_cos [THEN DERIV_unique]

             simp add: differentiable_def)

 apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])

 done

 lemma sin_total:

      "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"

 apply (rule ccontr)

 apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")

 apply (erule contrapos_np)

 apply (simp del: minus_sin_cos_eq [symmetric])

 apply (cut_tac y="-y" in cos_total, simp) apply simp

 apply (erule ex1E)

 apply (rule_tac a = "x - (pi/2)" in ex1I)

 apply (simp (no_asm) add: add_assoc)

 apply (rotate_tac 3)

 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)

 done

 lemma reals_Archimedean4:

      "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> 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 \<le> x; cos x = 0 |] ==>

       \<exists>n::nat. ~even n & x = real n * (pi/2)"

 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)

 apply (subgoal_tac "0 \<le> x - real n * pi &

                     (x - real n * pi) \<le> 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 \<le> x & x \<le> 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)

 done

 lemma sin_zero_lemma:

      "[| 0 \<le> x; sin x = 0 |] ==>

       \<exists>n::nat. even n & x = real n * (pi/2)"

 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")

  apply (clarify, rule_tac x = "n - 1" in exI)

  apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)

 apply (rule cos_zero_lemma)

 apply (simp_all add: add_increasing)

 done

 lemma cos_zero_iff:

      "(cos x = 0) =

       ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |

        (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"

 apply (rule iffI)

 apply (cut_tac linorder_linear [of 0 x], safe)

 apply (drule cos_zero_lemma, assumption+)

 apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)

 apply (force simp add: minus_equation_iff [of x])

 apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)

 apply (auto simp add: cos_add)

 done

 (* ditto: but to a lesser extent *)

 lemma sin_zero_iff:

      "(sin x = 0) =

       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |

        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"

 apply (rule iffI)

 apply (cut_tac linorder_linear [of 0 x], safe)

 apply (drule sin_zero_lemma, assumption+)

 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)

 apply (force simp add: minus_equation_iff [of x])

 apply (auto simp add: even_mult_two_ex)

 done

 lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> 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 auto

 qed

 lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"

 proof (cases "y < x")

   case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto

 next

   case False hence "y = x" using `y \<le> x` by auto

   thus ?thesis by auto

 qed

 lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"

   shows "cos y < cos x"

 proof -

   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto

   from cos_monotone_0_pi[OF this]

   show ?thesis unfolding cos_minus .

 qed

 lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"

 proof (cases "y < x")

   case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto

 next

   case False hence "y = x" using `y \<le> x` by auto

   thus ?thesis by auto

 qed

 lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"

 proof -

   have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> 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 auto

 qed

 subsection {* Tangent *}

 definition

   tan :: "real => real" where

   "tan x = (sin x)/(cos x)"

 lemma tan_zero [simp]: "tan 0 = 0"

 by (simp add: tan_def)

 lemma tan_pi [simp]: "tan pi = 0"

 by (simp add: tan_def)

 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"

 by (simp add: tan_def)

 lemma tan_minus [simp]: "tan (-x) = - tan x"

 by (simp add: tan_def minus_mult_left)

 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"

 by (simp add: tan_def)

 lemma lemma_tan_add1:

       "[| cos x \<noteq> 0; cos y \<noteq> 0 |]

         ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"

 apply (simp add: tan_def divide_inverse)

 apply (auto simp del: inverse_mult_distrib

             simp add: inverse_mult_distrib [symmetric] mult_ac)

 apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])

 apply (auto simp del: inverse_mult_distrib

             simp add: mult_assoc left_diff_distrib cos_add)

 done

 lemma add_tan_eq:

       "[| cos x \<noteq> 0; cos y \<noteq> 0 |]

        ==> tan x + tan y = sin(x + y)/(cos x * cos y)"

 apply (simp add: tan_def)

 apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])

 apply (auto simp add: mult_assoc left_distrib)

 apply (simp add: sin_add)

 done

 lemma tan_add:

      "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]

       ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"

 apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)

 apply (simp add: tan_def)

 done

 lemma tan_double:

      "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]

       ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"

 apply (insert tan_add [of x x])

 apply (simp add: mult_2 [symmetric])

 apply (auto simp add: numeral_2_eq_2)

 done

 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"

 by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)

 lemma tan_less_zero:

   assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"

 proof -

   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)

   thus ?thesis by simp

 qed

 lemma tan_half: fixes x :: real assumes "- (pi / 2) < x" and "x < pi / 2"

   shows "tan x = sin (2 * x) / (cos (2 * x) + 1)"

 proof -

   from cos_gt_zero_pi[OF `- (pi / 2) < x` `x < pi / 2`]

   have "cos x \<noteq> 0" by auto

   have minus_cos_2x: "\<And>X. X - cos (2*x) = X - (cos x) ^ 2 + (sin x) ^ 2" unfolding cos_double by algebra

   have "tan x = (tan x + tan x) / 2" by auto

   also have "\<dots> = sin (x + x) / (cos x * cos x) / 2" unfolding add_tan_eq[OF `cos x \<noteq> 0` `cos x \<noteq> 0`] ..

   also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + (cos x) ^ 2 + cos (2*x) - cos (2*x))" unfolding divide_divide_eq_left numeral_2_eq_2 by auto

   also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + cos (2*x) + (sin x)^2)" unfolding minus_cos_2x by auto

   also have "\<dots> = sin (2 * x) / (cos (2*x) + 1)" by auto

   finally show ?thesis .

 qed

 lemma lemma_DERIV_tan:

      "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"

   by (auto intro!: DERIV_intros simp add: field_simps numeral_2_eq_2)

 lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"

 by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])

 lemma isCont_tan [simp]: "cos x \<noteq> 0 ==> isCont tan x"

 by (rule DERIV_tan [THEN DERIV_isCont])

 lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"

 apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")

 apply (simp add: divide_inverse [symmetric])

 apply (rule LIM_mult)

 apply (rule_tac [2] inverse_1 [THEN subst])

 apply (rule_tac [2] LIM_inverse)

 apply (simp_all add: divide_inverse [symmetric])

 apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric])

 apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+

 done

 lemma lemma_tan_total: "0 < y ==> \<exists>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)

 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)

 done

 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"

 apply (frule order_le_imp_less_or_eq, safe)

  prefer 2 apply force

 apply (drule lemma_tan_total, safe)

 apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)

 apply (auto intro!: DERIV_tan [THEN DERIV_isCont])

 apply (drule_tac y = xa in order_le_imp_less_or_eq)

 apply (auto dest: cos_gt_zero)

 done

 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"

 apply (cut_tac linorder_linear [of 0 y], safe)

 apply (drule tan_total_pos)

 apply (cut_tac [2] y="-y" in tan_total_pos, safe)

 apply (rule_tac [3] x = "-x" in exI)

 apply (auto intro!: exI)

 done

 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"

 apply (cut_tac y = y in lemma_tan_total1, auto)

 apply (cut_tac x = xa and y = y in linorder_less_linear, auto)

 apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")

 apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")

 apply (rule_tac [4] Rolle)

 apply (rule_tac [2] Rolle)

 apply (auto intro!: DERIV_tan DERIV_isCont exI

             simp add: differentiable_def)

 txt{*Now, simulate TRYALL*}

 apply (rule_tac [!] DERIV_tan asm_rl)

 apply (auto dest!: DERIV_unique [OF _ DERIV_tan]

             simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])

 done

 lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"

   shows "tan y < tan x"

 proof -

   have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"

   proof (rule allI, rule impI)

     fix x' :: real assume "y \<le> x' \<and> x' \<le> x"

     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto

     from cos_gt_zero_pi[OF this]

     have "cos x' \<noteq> 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)\<twosuperior>)" 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)\<twosuperior>)" 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 auto

 qed

 lemma 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 auto

 next

   assume "tan y < tan x"

   show "y < x"

   proof (rule ccontr)

     assume "\<not> y < x" hence "x \<le> y" by auto

     hence "tan x \<le> tan y"

     proof (cases "x = y")

       case True thus ?thesis by auto

     next

       case False hence "x < y" using `x \<le> 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

   qed

 qed

 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto

 lemma 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 left_distrib by auto

   show ?case unfolding split_pi_off using Suc by auto

 qed auto

 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"

 proof (cases "0 \<le> i")

   case True hence i_nat: "real i = real (nat i)" by auto

   show ?thesis unfolding i_nat by auto

 next

   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 "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)

   finally show ?thesis by auto

 qed

 lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x"

   using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of .

 subsection {* Inverse Trigonometric Functions *}

 definition

   arcsin :: "real => real" where

   "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"

 definition

   arccos :: "real => real" where

   "arccos y = (THE x. 0 \<le> x & x \<le> 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 \<le> y; y \<le> 1 |]

       ==> -(pi/2) \<le> arcsin y &

            arcsin y \<le> pi/2 & sin(arcsin y) = y"

 unfolding arcsin_def by (rule theI' [OF sin_total])

 lemma arcsin_pi:

      "[| -1 \<le> y; y \<le> 1 |]

       ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"

 apply (drule (1) arcsin)

 apply (force intro: order_trans)

 done

 lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"

 by (blast dest: arcsin)

 lemma arcsin_bounded:

      "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"

 by (blast dest: arcsin)

 lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"

 by (blast dest: arcsin)

 lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"

 by (blast dest: arcsin)

 lemma arcsin_lt_bounded:

      "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"

 apply (frule order_less_imp_le)

 apply (frule_tac y = y in order_less_imp_le)

 apply (frule arcsin_bounded)

 apply (safe, simp)

 apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)

 apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)

 apply (drule_tac [!] f = sin in arg_cong, auto)

 done

 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"

 apply (unfold arcsin_def)

 apply (rule the1_equality)

 apply (rule sin_total, auto)

 done

 lemma arccos:

      "[| -1 \<le> y; y \<le> 1 |]

       ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"

 unfolding arccos_def by (rule theI' [OF cos_total])

 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"

 by (blast dest: arccos)

 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"

 by (blast dest: arccos)

 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"

 by (blast dest: arccos)

 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"

 by (blast dest: arccos)

 lemma arccos_lt_bounded:

      "[| -1 < y; y < 1 |]

       ==> 0 < arccos y & arccos y < pi"

 apply (frule order_less_imp_le)

 apply (frule_tac y = y in order_less_imp_le)

 apply (frule arccos_bounded, auto)

 apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)

 apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)

 apply (drule_tac [!] f = cos in arg_cong, auto)

 done

 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"

 apply (simp add: arccos_def)

 apply (auto intro!: the1_equality cos_total)

 done

 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"

 apply (simp add: arccos_def)

 apply (auto intro!: the1_equality cos_total)

 done

 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"

 apply (subgoal_tac "x\<twosuperior> \<le> 1")

 apply (rule power2_eq_imp_eq)

 apply (simp add: cos_squared_eq)

 apply (rule cos_ge_zero)

 apply (erule (1) arcsin_lbound)

 apply (erule (1) arcsin_ubound)

 apply simp

 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)

 apply (rule power_mono, simp, simp)

 done

 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"

 apply (subgoal_tac "x\<twosuperior> \<le> 1")

 apply (rule power2_eq_imp_eq)

 apply (simp add: sin_squared_eq)

 apply (rule sin_ge_zero)

 apply (erule (1) arccos_lbound)

 apply (erule (1) arccos_ubound)

 apply simp

 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)

 apply (rule power_mono, simp, simp)

 done

 lemma arctan [simp]:

      "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"

 unfolding arctan_def by (rule theI' [OF tan_total])

 lemma tan_arctan: "tan(arctan y) = y"

 by auto

 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"

 by (auto simp only: arctan)

 lemma arctan_lbound: "- (pi/2) < arctan y"

 by auto

 lemma arctan_ubound: "arctan y < pi/2"

 by (auto simp only: arctan)

 lemma arctan_tan:

       "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"

 apply (unfold arctan_def)

 apply (rule the1_equality)

 apply (rule tan_total, auto)

 done

 lemma arctan_zero_zero [simp]: "arctan 0 = 0"

 by (insert arctan_tan [of 0], simp)

 lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"

 apply (auto simp add: cos_zero_iff)

 apply (case_tac "n")

 apply (case_tac [3] "n")

 apply (cut_tac [2] y = x in arctan_ubound)

 apply (cut_tac [4] y = x in arctan_lbound)

 apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)

 done

 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"

 apply (rule power_inverse [THEN subst])

 apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])

 apply (auto dest: field_power_not_zero

         simp add: power_mult_distrib left_distrib power_divide tan_def

                   mult_assoc power_inverse [symmetric])

 done

 lemma isCont_inverse_function2:

   fixes f g :: "real \<Rightarrow> real" shows

   "\<lbrakk>a < x; x < b;

     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;

     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>

    \<Longrightarrow> isCont g (f x)"

 apply (rule isCont_inverse_function

        [where f=f and d="min (x - a) (b - x)"])

 apply (simp_all add: abs_le_iff)

 done

 lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"

 apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)

 apply (rule isCont_inverse_function2 [where f=sin])

 apply (erule (1) arcsin_lt_bounded [THEN conjunct1])

 apply (erule (1) arcsin_lt_bounded [THEN conjunct2])

 apply (fast intro: arcsin_sin, simp)

 done

 lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"

 apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)

 apply (rule isCont_inverse_function2 [where f=cos])

 apply (erule (1) arccos_lt_bounded [THEN conjunct1])

 apply (erule (1) arccos_lt_bounded [THEN conjunct2])

 apply (fast intro: arccos_cos, simp)

 done

 lemma isCont_arctan: "isCont arctan x"

 apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)

 apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)

 apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)

 apply (erule (1) isCont_inverse_function2 [where f=tan])

 apply (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)

 done

 lemma DERIV_arcsin:

   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"

 apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])

 apply (rule lemma_DERIV_subst [OF DERIV_sin])

 apply (simp add: cos_arcsin)

 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)

 apply (rule power_strict_mono, simp, simp, simp)

 apply assumption

 apply assumption

 apply simp

 apply (erule (1) isCont_arcsin)

 done

 lemma DERIV_arccos:

   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<twosuperior>))"

 apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])

 apply (rule lemma_DERIV_subst [OF DERIV_cos])

 apply (simp add: sin_arccos)

 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)

 apply (rule power_strict_mono, simp, simp, simp)

 apply assumption

 apply assumption

 apply simp

 apply (erule (1) isCont_arccos)

 done

 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<twosuperior>)"

 apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])

 apply (rule lemma_DERIV_subst [OF DERIV_tan])

 apply (rule cos_arctan_not_zero)

 apply (simp add: power_inverse tan_sec [symmetric])

 apply (subgoal_tac "0 < 1 + x\<twosuperior>", simp)

 apply (simp add: add_pos_nonneg)

 apply (simp, simp, simp, rule isCont_arctan)

 done

 declare

   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]

 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 \<le> ?c"

     by (rule cos_ge_zero, rule order_trans [where y=0], simp_all)

   have "0 = cos (pi / 4 + pi / 4)"

     by simp

   also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"

     by (simp only: cos_add power2_eq_square)

   also have "\<dots> = 2 * ?c\<twosuperior> - 1"

     by (simp add: sin_squared_eq)

   finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"

     by (simp add: power_divide)

   thus ?thesis

     using nonneg by (rule power2_eq_imp_eq) simp

 qed

 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"

 proof -

   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"

   have pos_c: "0 < ?c"

     by (rule cos_gt_zero, simp, simp)

   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"

     by simp

   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"

     by (simp only: cos_add sin_add)

   also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"

     by (simp add: algebra_simps power2_eq_square)

   finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"

     using pos_c by (simp add: sin_squared_eq power_divide)

   thus ?thesis

     using pos_c [THEN order_less_imp_le]

     by (rule power2_eq_imp_eq) simp

 qed

 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"

 proof -

   have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq)

   also have "pi / 2 - pi / 4 = pi / 4" by simp

   also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45)

   finally show ?thesis .

 qed

 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"

 proof -

   have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq)

   also have "pi / 2 - pi / 3 = pi / 6" by simp

   also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30)

   finally show ?thesis .

 qed

 lemma cos_60: "cos (pi / 3) = 1 / 2"

 apply (rule power2_eq_imp_eq)

 apply (simp add: cos_squared_eq sin_60 power_divide)

 apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)

 done

 lemma sin_30: "sin (pi / 6) = 1 / 2"

 proof -

   have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq)

   also have "pi / 2 - pi / 6 = pi / 3" by simp

   also have "cos (pi / 3) = 1 / 2" by (rule cos_60)

   finally show ?thesis .

 qed

 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"

 unfolding tan_def by (simp add: sin_30 cos_30)

 lemma tan_45: "tan (pi / 4) = 1"

 unfolding tan_def by (simp add: sin_45 cos_45)

 lemma tan_60: "tan (pi / 3) = sqrt 3"

 unfolding tan_def by (simp add: sin_60 cos_60)

 lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"

   by (auto intro!: DERIV_intros)

 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 left_distrib add_divide_distrib

                   mult_commute [of pi])

 qed

 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"

 by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)

 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"

 apply (subgoal_tac "cos (pi + pi/2) = 0", simp)

 apply (subst cos_add, simp)

 done

 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"

 by (auto simp add: mult_assoc)

 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"

 apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)

 apply (subst sin_add, simp)

 done

 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"

 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)

 lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"

   by (auto intro!: DERIV_intros)

 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 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 tan_total_pi4: assumes "\<bar>x\<bar> < 1"

   shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"

 proof -

   obtain z where "- (pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast

   have "tan (pi / 4) = 1" and "tan (- (pi / 4)) = - 1" using tan_45 tan_minus by auto

   have "z \<noteq> pi / 4"

   proof (rule ccontr)

     assume "\<not> (z \<noteq> pi / 4)" hence "z = pi / 4" by auto

     have "tan z = 1" unfolding `z = pi / 4` `tan (pi / 4) = 1` ..

     thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto

   qed

   have "z \<noteq> - (pi / 4)"

   proof (rule ccontr)

     assume "\<not> (z \<noteq> - (pi / 4))" hence "z = - (pi / 4)" by auto

     have "tan z = - 1" unfolding `z = - (pi / 4)` `tan (- (pi / 4)) = - 1` ..

     thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto

   qed

   have "z < pi / 4"

   proof (rule ccontr)

     assume "\<not> (z < pi / 4)" hence "pi / 4 < z" using `z \<noteq> pi / 4` by auto

     have "- (pi / 2) < pi / 4" using m2pi_less_pi by auto

     from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`]

     have "1 < x" unfolding `tan z = x` `tan (pi / 4) = 1` .

     thus False using `\<bar>x\<bar> < 1` by auto

   qed

   moreover

   have "-(pi / 4) < z"

   proof (rule ccontr)

     assume "\<not> (-(pi / 4) < z)" hence "z < - (pi / 4)" using `z \<noteq> - (pi / 4)` by auto

     have "-(pi / 4) < pi / 2" using m2pi_less_pi by auto

     from tan_monotone[OF `-(pi / 2) < z` `z < -(pi / 4)` this]

     have "x < - 1" unfolding `tan z = x` `tan (-(pi / 4)) = - 1` .

     thus False using `\<bar>x\<bar> < 1` by auto

   qed

   ultimately show ?thesis using `tan z = x` by auto

 qed

 lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"

   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"

 proof -