(*  Title       : Transcendental.thy

    Author      : Jacques D. Fleuriot

    Copyright   : 1998,1999 University of Cambridge

                  1999,2001 University of Edinburgh

    Conversion to Isar and new proofs by Lawrence C Paulson, 2004

*)

header{*Power Series, Transcendental Functions etc.*}

theory Transcendental

imports NthRoot Fact Series EvenOdd Deriv

begin

subsection{*Properties of Power Series*}

lemma lemma_realpow_diff:

  fixes y :: "'a::recpower"

  shows "p \<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_Suc power_commutes)

qed

lemma lemma_realpow_diff_sumr:

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

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

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

by (auto simp add: setsum_right_distrib lemma_realpow_diff mult_ac

  simp del: setsum_op_ivl_Suc cong: strong_setsum_cong)

lemma lemma_realpow_diff_sumr2:

  fixes y :: "'a::{recpower,comm_ring}" shows

     "x ^ (Suc n) - y ^ (Suc n) =  

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

apply (induct "n", simp add: power_Suc)

apply (simp add: power_Suc del: setsum_op_ivl_Suc)

apply (subst setsum_op_ivl_Suc)

apply (subst lemma_realpow_diff_sumr)

apply (simp add: right_distrib del: setsum_op_ivl_Suc)

apply (subst mult_left_commute [where a="x - y"])

apply (erule subst)

apply (simp add: power_Suc ring_eq_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,recpower}"

  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,recpower}" shows

     "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]  

      ==> summable (%n. f(n) * (z ^ n))"

by (rule powser_insidea [THEN summable_norm_cancel])

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

definition

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

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

text{*Lemma about distributing negation over it*}

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

by (simp add: diffs_def)

text{*Show that we can shift the terms down one*}

lemma lemma_diffs:

     "(\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) =  

      (\<Sum>n=0..<n. of_nat n * c(n) * (x ^ (n - Suc 0))) +  

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

apply (induct "n")

apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)

done

lemma lemma_diffs2:

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

      (\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) -  

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

by (auto simp add: lemma_diffs)

lemma diffs_equiv:

     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  

      (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums  

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

apply (subgoal_tac " (%n. of_nat n * c (n) * (x ^ (n - Suc 0))) ----> 0")

apply (rule_tac [2] LIMSEQ_imp_Suc)

apply (drule summable_sums) 

apply (auto simp add: sums_def)

apply (drule_tac X="(\<lambda>n. \<Sum>n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff)

apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])

apply (simp add: diffs_def summable_LIMSEQ_zero)

done

lemma lemma_termdiff1:

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

  "(\<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: right_distrib diff_minus power_add [symmetric] mult_ac

  cong: strong_setsum_cong)

lemma less_add_one: "m < n ==> (\<exists>d. n = m + d + Suc 0)"

by (simp add: less_iff_Suc_add)

lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"

by arith

lemma sumr_diff_mult_const2:

  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"

by (simp add: setsum_subtractf)

lemma lemma_termdiff2:

  fixes h :: "'a :: {recpower,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 only: times_divide_eq_left [symmetric])

apply (simp add: divide_self [OF h])

apply (simp add: mult_assoc [symmetric])

apply (cases "n", simp)

apply (simp add: lemma_realpow_diff_sumr2 h

                 right_diff_distrib [symmetric] mult_assoc

            del: realpow_Suc setsum_op_ivl_Suc of_nat_Suc)

apply (subst lemma_realpow_rev_sumr)

apply (subst sumr_diff_mult_const2)

apply simp

apply (simp only: lemma_termdiff1 setsum_right_distrib)

apply (rule setsum_cong [OF refl])

apply (simp add: diff_minus [symmetric] less_iff_Suc_add)

apply (clarify)

apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac

            del: setsum_op_ivl_Suc realpow_Suc)

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

apply (simp add: mult_ac)

done

lemma real_setsum_nat_ivl_bounded2:

  fixes K :: "'a::ordered_semidom"

  assumes f: "\<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,recpower}"

  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,recpower} \<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"

proof (simp add: LIM_def, safe)

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

  have zero_le_K: "0 \<le> K"

    apply (cut_tac k)

    apply (cut_tac h="of_real (k/2)" in le, simp)

    apply (simp del: of_real_divide)

    apply (drule order_trans [OF norm_ge_zero])

    apply (simp add: zero_le_mult_iff)

    done

  show "\<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::{recpower,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::{recpower,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::{recpower,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)"

proof (simp add: deriv_def, 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" by (simp add: less_diff_eq 4)

  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_tac f="suminf" in arg_cong)

      apply (rule ext)

      apply (simp add: ring_eq_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{*Exponential Function*}

definition

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

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

definition

  sin :: "real => real" where

  "sin x = (\<Sum>n. (if even(n) then 0 else

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

definition

  cos :: "real => real" where

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

                            else 0) * x ^ n)"

lemma summable_exp_generic:

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

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

  shows "summable S"

proof -

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

    unfolding S_def by (simp add: power_Suc del: mult_Suc)

  obtain r :: real where r0: "0 < r" and r1: "r < 1"

    using dense [OF zero_less_one] by fast

  obtain N :: nat where N: "norm x < real N * r"

    using reals_Archimedean3 [OF r0] by fast

  from r1 show ?thesis

  proof (rule ratio_test [rule_format])

    fix n :: nat

    assume n: "N \<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 norm_scaleR inverse_eq_divide)

  qed

qed

lemma summable_norm_exp:

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

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

proof (rule summable_norm_comparison_test [OF exI, rule_format])

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

    by (rule summable_exp_generic)

next

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

    by (simp add: norm_scaleR norm_power_ineq)

qed

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

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

lemma summable_sin: 

     "summable (%n.  

           (if even n then 0  

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

                x ^ n)"

apply (rule_tac g = "(%n. inverse (real (fact n)) * \<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 (%n.  

           (if even n then  

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

apply (rule_tac g = "(%n. inverse (real (fact n)) * \<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 exp_converges: "(\<lambda>n. x ^ n /# real (fact n)) sums exp x"

unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])

lemma sin_converges: 

      "(%n. (if even n then 0  

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

                 x ^ n) sums sin(x)"

unfolding sin_def by (rule summable_sin [THEN summable_sums])

lemma cos_converges: 

      "(%n. (if even n then  

           -1 ^ (n div 2)/(real (fact n))  

           else 0) * x ^ n) sums cos(x)"

unfolding cos_def by (rule summable_cos [THEN summable_sums])

subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 

lemma exp_fdiffs: 

      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"

by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def

         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 sin_fdiffs: 

      "diffs(%n. if even n then 0  

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

       = (%n. if even n then  

                 -1 ^ (n div 2)/(real (fact n))  

              else 0)"

by (auto intro!: ext 

         simp add: diffs_def divide_inverse real_of_nat_def

         simp del: mult_Suc of_nat_Suc)

lemma sin_fdiffs2: 

       "diffs(%n. if even n then 0  

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

       = (if even n then  

                 -1 ^ (n div 2)/(real (fact n))  

              else 0)"

by (simp only: sin_fdiffs)

lemma cos_fdiffs: 

      "diffs(%n. if even n then  

                 -1 ^ (n div 2)/(real (fact n)) else 0)  

       = (%n. - (if even n then 0  

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

by (auto intro!: ext 

         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def

         simp del: mult_Suc of_nat_Suc)

lemma cos_fdiffs2: 

      "diffs(%n. if even n then  

                 -1 ^ (n div 2)/(real (fact n)) else 0) n 

       = - (if even n then 0  

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

by (simp only: cos_fdiffs)

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

lemma lemma_sin_minus:

     "- sin x = (\<Sum>n. - ((if even n then 0 

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

by (auto intro!: sums_unique sums_minus sin_converges)

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

by (auto intro!: ext simp add: exp_def)

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

apply (simp add: exp_def)

apply (subst lemma_exp_ext)

apply (subgoal_tac "DERIV (\<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 lemma_sin_ext:

     "sin = (%x. \<Sum>n. 

                   (if even n then 0  

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

                   x ^ n)"

by (auto intro!: ext simp add: sin_def)

lemma lemma_cos_ext:

     "cos = (%x. \<Sum>n. 

                   (if even n then -1 ^ (n div 2)/(real (fact n)) else 0) *

                   x ^ n)"

by (auto intro!: ext simp add: cos_def)

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

apply (simp add: cos_def)

apply (subst lemma_sin_ext)

apply (auto simp add: sin_fdiffs2 [symmetric])

apply (rule_tac K = "1 + \<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_exp [simp]: "isCont exp x"

by (rule DERIV_exp [THEN DERIV_isCont])

lemma isCont_sin [simp]: "isCont sin x"

by (rule DERIV_sin [THEN DERIV_isCont])

lemma isCont_cos [simp]: "isCont cos x"

by (rule DERIV_cos [THEN DERIV_isCont])

subsection{*Properties of the Exponential Function*}

lemma powser_zero:

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

  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 by simp

qed

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

unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)

lemma setsum_head2:

  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"

by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

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_head2 setsum_shift_bounds_cl_Suc_ivl

         del: setsum_cl_ivl_Suc)

lemma exp_series_add:

  fixes x y :: "'a::{real_field,recpower}"

  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /# 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) /# real (Suc n)"

    unfolding S_def by (simp add: power_Suc del: mult_Suc)

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

    by simp

  have "real (Suc n) *# 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) *# (S x (Suc i) * S y (n-i)))

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

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

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

    by (subst setsum_cl_ivl_Suc, simp)

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

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

             (\<Sum>i=0..Suc n. real (Suc n) *# (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) *# (\<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 add: scaleR_cancel_left del: setsum_cl_ivl_Suc)

qed

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

unfolding exp_def

by (simp only: Cauchy_product summable_norm_exp exp_series_add)

lemma exp_of_real: "exp (of_real x) = of_real (exp x)"

unfolding exp_def

apply (subst of_real.suminf)

apply (rule summable_exp_generic)

apply (simp add: scaleR_conv_of_real)

done

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

apply (drule order_le_imp_less_or_eq, auto)

apply (simp add: exp_def)

apply (rule real_le_trans)

apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)

apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)

done

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

apply (rule order_less_le_trans)

apply (rule_tac [2] exp_ge_add_one_self_aux, auto)

done

lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"

proof -

  have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"

    by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_ident DERIV_const) 

  thus ?thesis by (simp add: o_def)

qed

lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"

proof -

  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"

    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_ident)

  thus ?thesis by (simp add: o_def)

qed

lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"

proof -

  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x

       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"

    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 

  thus ?thesis by (simp add: mult_commute)

qed

lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y::real)"

proof -

  have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp

  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 

    by (rule DERIV_isconst_all) 

  thus ?thesis by simp

qed

lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"

by (simp add: exp_add [symmetric])

lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"

by (simp add: mult_commute)

lemma exp_minus: "exp(-x) = inverse(exp(x))"

by (auto intro: inverse_unique [symmetric])

text{*Proof: because every exponential can be seen as a square.*}

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

apply (rule_tac t = x in real_sum_of_halves [THEN subst])

apply (subst exp_add, auto)

done

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

apply (cut_tac x = x in exp_mult_minus2)

apply (auto simp del: exp_mult_minus2)

done

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

by (simp add: order_less_le)

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

by (auto intro: positive_imp_inverse_positive)

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

by auto

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

apply (induct "n")

apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)

done

lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"

apply (simp add: diff_minus divide_inverse)

apply (simp (no_asm) add: exp_add exp_minus)

done

lemma exp_less_mono:

  fixes x y :: real

  assumes xy: "x < y" shows "exp x < exp y"

proof -

  have "1 < exp (y + - x)"

    by (rule real_less_sum_gt_zero [THEN exp_gt_one])

  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"

    by (auto simp add: exp_add exp_minus)

  thus ?thesis

    by (simp add: divide_inverse [symmetric] pos_less_divide_eq

             del: divide_self_if)

qed

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

apply (simp add: linorder_not_le [symmetric]) 

apply (auto simp add: order_le_less exp_less_mono) 

done

lemma exp_less_cancel_iff [iff]: "(exp(x::real) < exp(y)) = (x < y)"

by (auto intro: exp_less_mono exp_less_cancel)

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

by (auto simp add: linorder_not_less [symmetric])

lemma exp_inj_iff [iff]: "(exp (x::real) = exp y) = (x = y)"

by (simp add: order_eq_iff)

lemma lemma_exp_total: "1 \<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] real_inverse_gt_one)

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

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

apply (simp add: exp_minus)

done

subsection{*Properties of the Logarithmic Function*}

definition

  ln :: "real => real" where

  "ln x = (THE u. exp u = x)"

lemma ln_exp [simp]: "ln (exp x) = x"

by (simp add: ln_def)

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

by (auto dest: exp_total)

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

apply (auto dest: exp_total)

apply (erule subst, simp) 

done

lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"

apply (rule exp_inj_iff [THEN iffD1])

apply (simp add: exp_add exp_ln mult_pos_pos)

done

lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"

apply (simp only: exp_ln_iff [symmetric])

apply (erule subst)+

apply simp 

done

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

by (rule exp_inj_iff [THEN iffD1], auto)

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

apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])

apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])

done

lemma ln_div: 

    "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"

apply (simp add: divide_inverse)

apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)

done

lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"

apply (simp only: exp_ln_iff [symmetric])

apply (erule subst)+

apply simp 

done

lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"

by (auto simp add: linorder_not_less [symmetric])

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

by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])

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

apply (rule ln_exp [THEN subst])

apply (rule ln_le_cancel_iff [THEN iffD2]) 

apply (auto simp add: exp_ge_add_one_self_aux)

done

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

apply (rule order_less_le_trans)

apply (rule_tac [2] ln_add_one_self_le_self)

apply (rule ln_less_cancel_iff [THEN iffD2], auto)

done

lemma ln_ge_zero [simp]:

  assumes x: "1 \<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