src/HOL/Hyperreal/Transcendental.thy

suminf -> \<Sum>

(*  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 HSeries EvenOdd Lim
begin

constdefs
    root :: "[nat,real] => real"
    "root n x == (@u. ((0::real) < x --> 0 < u) & (u ^ n = x))"

    sqrt :: "real => real"
    "sqrt x == root 2 x"

    exp :: "real => real"
    "exp x == \<Sum>n. inverse(real (fact n)) * (x ^ n)"

    sin :: "real => real"
    "sin x == \<Sum>n. (if even(n) then 0 else
             ((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n"
 
    diffs :: "(nat => real) => nat => real"
    "diffs c == (%n. real (Suc n) * c(Suc n))"

    cos :: "real => real"
    "cos x == \<Sum>n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n)) 
                          else 0) * x ^ n"
  
    ln :: "real => real"
    "ln x == (@u. exp u = x)"

    pi :: "real"
    "pi == 2 * (@x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"

    tan :: "real => real"
    "tan x == (sin x)/(cos x)"

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

    arcos :: "real => real"
    "arcos y == (@x. 0 \<le> x & x \<le> pi & cos x = y)"
     
    arctan :: "real => real"
    "arctan y == (@x. -(pi/2) < x & x < pi/2 & tan x = y)"


lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
apply (simp add: root_def)
apply (safe intro!: some_equality power_0_Suc elim!: realpow_zero_zero)
done

lemma real_root_pow_pos: 
     "0 < x ==> (root(Suc n) x) ^ (Suc n) = x"
apply (simp add: root_def)
apply (drule_tac n = n in realpow_pos_nth2)
apply (auto intro: someI2)
done

lemma real_root_pow_pos2: "0 \<le> x ==> (root(Suc n) x) ^ (Suc n) = x"
by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos)

lemma real_root_pos: 
     "0 < x ==> root(Suc n) (x ^ (Suc n)) = x"
apply (simp add: root_def)
apply (rule some_equality)
apply (frule_tac [2] n = n in zero_less_power)
apply (auto simp add: zero_less_mult_iff)
apply (rule_tac x = u and y = x in linorder_cases)
apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less])
apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less])
apply (auto)
done

lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x"
by (auto dest!: real_le_imp_less_or_eq real_root_pos)

lemma real_root_pos_pos: 
     "0 < x ==> 0 \<le> root(Suc n) x"
apply (simp add: root_def)
apply (drule_tac n = n in realpow_pos_nth2)
apply (safe, rule someI2)
apply (auto intro!: order_less_imp_le dest: zero_less_power 
            simp add: zero_less_mult_iff)
done

lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x"
by (auto dest!: real_le_imp_less_or_eq dest: real_root_pos_pos)

lemma real_root_one [simp]: "root (Suc n) 1 = 1"
apply (simp add: root_def)
apply (rule some_equality, auto)
apply (rule ccontr)
apply (rule_tac x = u and y = 1 in linorder_cases)
apply (drule_tac n = n in realpow_Suc_less_one)
apply (drule_tac [4] n = n in power_gt1_lemma)
apply (auto)
done


subsection{*Square Root*}

text{*needed because 2 is a binary numeral!*}
lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"
by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 
         add: nat_numeral_0_eq_0 [symmetric])

lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
by (simp add: sqrt_def)

lemma real_sqrt_one [simp]: "sqrt 1 = 1"
by (simp add: sqrt_def)

lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
apply (simp add: sqrt_def)
apply (rule iffI) 
 apply (cut_tac r = "root 2 x" in realpow_two_le)
 apply (simp add: numeral_2_eq_2)
apply (subst numeral_2_eq_2)
apply (erule real_root_pow_pos2)
done

lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>"
by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])

lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x"
by (simp)

lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>"
by (rule real_sqrt_pow2_iff [THEN iffD2], arith)

lemma real_pow_sqrt_eq_sqrt_pow: 
      "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)"
apply (simp add: sqrt_def)
apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2)
done

lemma real_pow_sqrt_eq_sqrt_abs_pow2:
     "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])

lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>"
apply (rule real_sqrt_abs_abs [THEN subst])
apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst])
apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric])
apply (assumption, arith)
done

lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
apply auto
apply (cut_tac x = x and y = 0 in linorder_less_linear)
apply (simp add: zero_less_mult_iff)
done

lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"
apply (simp add: sqrt_def root_def)
apply (drule realpow_pos_nth2 [where n=1], safe)
apply (rule someI2, auto)
done

lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
by (auto intro: real_sqrt_gt_zero simp add: order_le_less)

lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 


(*we need to prove something like this:
lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r"
apply (case_tac n, simp) 
apply (simp add: root_def) 
apply (rule someI2 [of _ r], safe)
apply (auto simp del: realpow_Suc dest: power_inject_base)
*)

lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r"
apply (unfold sqrt_def root_def) 
apply (rule someI2 [of _ r], auto) 
apply (auto simp add: numeral_2_eq_2 simp del: realpow_Suc 
            dest: power_inject_base) 
done

lemma real_sqrt_mult_distrib: 
     "[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)"
apply (rule sqrt_eqI)
apply (simp add: power_mult_distrib)  
apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
done

lemma real_sqrt_mult_distrib2:
     "[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) =  sqrt(x) * sqrt(y)"
by (auto intro: real_sqrt_mult_distrib simp add: order_le_less)

lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
by (auto intro!: real_sqrt_ge_zero)

lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
     "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)

lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
     "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
by (auto simp add: zero_le_mult_iff simp del: realpow_Suc)

lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>"
apply (rule abs_realpow_two [THEN subst])
apply (rule real_sqrt_abs_abs [THEN subst])
apply (subst real_pow_sqrt_eq_sqrt_pow)
apply (auto simp add: numeral_2_eq_2)
done

lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
apply (rule realpow_two [THEN subst])
apply (subst numeral_2_eq_2 [symmetric])
apply (rule real_sqrt_abs)
done

lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
by simp

lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
apply (frule real_sqrt_pow2_gt_zero)
apply (auto simp add: numeral_2_eq_2)
done

lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
by (cut_tac n1 = 2 and a1 = "sqrt x" in power_inverse [symmetric], auto)

lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
apply (drule real_le_imp_less_or_eq)
apply (auto dest: real_sqrt_not_eq_zero)
done

lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))"
by (auto simp add: real_sqrt_eq_zero_cancel)

lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe)
apply (rule real_le_trans)
apply (auto simp del: realpow_Suc)
apply (rule_tac n = 1 in realpow_increasing)
apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc)
done

lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)"
apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
done

lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
apply (rule_tac n = 1 in realpow_increasing)
apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp 
            del: realpow_Suc)
done


subsection{*Exponential Function*}

lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
apply (cut_tac 'a = real in zero_less_one [THEN dense], safe)
apply (cut_tac x = r in reals_Archimedean3, auto)
apply (drule_tac x = "\<bar>x\<bar>" in spec, safe)
apply (rule_tac N = n and c = r in ratio_test)
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
apply (rule mult_right_mono)
apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
apply (subst fact_Suc)
apply (subst real_of_nat_mult)
apply (auto)
apply (auto simp add: mult_assoc [symmetric] positive_imp_inverse_positive)
apply (rule order_less_imp_le)
apply (rule_tac z1 = "real (Suc na)" in real_mult_less_iff1 [THEN iffD1])
apply (auto simp add: real_not_refl2 [THEN not_sym] mult_assoc)
apply (erule order_less_trans)
apply (auto simp add: mult_less_cancel_left mult_ac)
done

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 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 power_abs [symmetric] zero_le_mult_iff)
done

lemma lemma_STAR_sin [simp]:
     "(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 [simp]:
     "0 < n -->  
      (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
by (induct "n", auto)

lemma lemma_STAR_cos1 [simp]:
     "0 < n -->  
      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
by (induct "n", auto)

lemma lemma_STAR_cos2 [simp]:
  "(\<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: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
apply (simp add: exp_def)
apply (rule summable_exp [THEN summable_sums])
done

lemma sin_converges: 
      "(%n. (if even n then 0  
            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
                 x ^ n) sums sin(x)"
apply (simp add: sin_def)
apply (rule summable_sin [THEN summable_sums])
done

lemma cos_converges: 
      "(%n. (if even n then  
           (- 1) ^ (n div 2)/(real (fact n))  
           else 0) * x ^ n) sums cos(x)"
apply (simp add: cos_def)
apply (rule summable_cos [THEN summable_sums])
done

lemma lemma_realpow_diff [rule_format (no_asm)]:
     "p \<le> n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y"
apply (induct "n", auto)
apply (subgoal_tac "p = Suc n")
apply (simp (no_asm_simp), auto)
apply (drule sym)
apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric] 
       del: realpow_Suc)
done


subsection{*Properties of Power Series*}

lemma lemma_realpow_diff_sumr:
     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ ((Suc n) - p)) =  
      y * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))::real)"
apply (auto simp add: setsum_mult simp del: setsum_Suc)
apply (rule setsum_cong[OF refl])
apply (subst lemma_realpow_diff)
apply (auto simp add: mult_ac)
done

lemma lemma_realpow_diff_sumr2:
     "x ^ (Suc n) - y ^ (Suc n) =  
      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^(n - p))::real)"
apply (induct "n", simp)
apply (auto simp del: setsum_Suc)
apply (subst setsum_Suc)
apply (drule sym)
apply (auto simp add: lemma_realpow_diff_sumr right_distrib diff_minus mult_ac simp del: setsum_Suc)
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)::real)"
apply (case_tac "x = y")
apply (auto simp add: mult_commute power_add [symmetric] simp del: setsum_Suc)
apply (rule_tac c1 = "x - y" in real_mult_left_cancel [THEN iffD1])
apply (rule_tac [2] minus_minus [THEN subst], simp)
apply (subst minus_mult_left)
apply (simp add: lemma_realpow_diff_sumr2 [symmetric] del: setsum_Suc)
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:
     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
      ==> summable (%n. \<bar>f(n)\<bar> * (z ^ n))"
apply (drule summable_LIMSEQ_zero)
apply (drule convergentI)
apply (simp add: Cauchy_convergent_iff [symmetric])
apply (drule Cauchy_Bseq)
apply (simp add: Bseq_def, safe)
apply (rule_tac g = "%n. K * \<bar>z ^ n\<bar> * inverse (\<bar>x ^ n\<bar>)" in summable_comparison_test)
apply (rule_tac x = 0 in exI, safe)
apply (subgoal_tac "0 < \<bar>x ^ n\<bar> ")
apply (rule_tac c="\<bar>x ^ n\<bar>" in mult_right_le_imp_le) 
apply (auto simp add: mult_assoc power_abs)
 prefer 2
 apply (drule_tac x = 0 in spec, force)
apply (auto simp add: power_abs mult_ac)
apply (rule_tac a2 = "z ^ n" 
       in abs_ge_zero [THEN real_le_imp_less_or_eq, THEN disjE])
apply (auto intro!: mult_right_mono simp add: mult_assoc [symmetric] power_abs summable_def power_0_left)
apply (rule_tac x = "K * inverse (1 - (\<bar>z\<bar> * inverse (\<bar>x\<bar>)))" in exI)
apply (auto intro!: sums_mult simp add: mult_assoc)
apply (subgoal_tac "\<bar>z ^ n\<bar> * inverse (\<bar>x\<bar> ^ n) = (\<bar>z\<bar> * inverse (\<bar>x\<bar>)) ^ n")
apply (auto simp add: power_abs [symmetric])
apply (subgoal_tac "x \<noteq> 0")
apply (subgoal_tac [3] "x \<noteq> 0")
apply (auto simp del: abs_inverse abs_mult simp add: abs_inverse [symmetric] realpow_not_zero abs_mult [symmetric] power_inverse power_mult_distrib [symmetric])
apply (auto intro!: geometric_sums  simp add: power_abs inverse_eq_divide)
done

lemma powser_inside:
     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
      ==> summable (%n. f(n) * (z ^ n))"
apply (drule_tac z = "\<bar>z\<bar>" in powser_insidea)
apply (auto intro: summable_rabs_cancel simp add: power_abs [symmetric])
done


subsection{*Differentiation of Power Series*}

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. real n * c(n) * (x ^ (n - Suc 0))) +  
      (real 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. real n * c(n) * (x ^ (n - Suc 0))) =  
      (\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) -  
      (real n * c(n) * x ^ (n - Suc 0))"
by (auto simp add: lemma_diffs)


lemma diffs_equiv:
     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
      (%n. real n * c(n) * (x ^ (n - Suc 0))) sums  
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
apply (subgoal_tac " (%n. real 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


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

lemma lemma_termdiff1:
  "(\<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)))::real)"
apply (rule setsum_cong[OF refl])
apply (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac)
done

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 lemma_termdiff2:
  "h \<noteq> 0 ==>
   (((z + h) ^ n) - (z ^ n)) * inverse h - real n * (z ^ (n - Suc 0)) =
   h * (\<Sum>p=0..< n - Suc 0. (z ^ p) *
       (\<Sum>q=0..< (n - Suc 0) - p. ((z + h) ^ q) * (z ^ (((n - 2) - p) - q))))"
apply (rule real_mult_left_cancel [THEN iffD1], simp (no_asm_simp))
apply (simp add: right_diff_distrib mult_ac)
apply (simp add: mult_assoc [symmetric])
apply (case_tac "n")
apply (auto simp add: lemma_realpow_diff_sumr2 
                      right_diff_distrib [symmetric] mult_assoc
            simp del: realpow_Suc setsum_Suc)
apply (auto simp add: lemma_realpow_rev_sumr simp del: setsum_Suc)
apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 
                sumdiff lemma_termdiff1 setsum_mult)
apply (auto intro!: setsum_cong[OF refl] simp add: diff_minus real_add_assoc)
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
apply (auto simp add: setsum_mult lemma_realpow_diff_sumr2 mult_ac simp
                 del: setsum_Suc realpow_Suc)
done

lemma lemma_termdiff3:
     "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; \<bar>z + h\<bar> \<le> K |]  
      ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))  
          \<le> real n * real (n - Suc 0) * K ^ (n - 2) * \<bar>h\<bar>"
apply (subst lemma_termdiff2, assumption)
apply (simp add: mult_commute) 
apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 
apply (rule setsum_abs [THEN real_le_trans])
apply (simp add: mult_assoc [symmetric])
apply (simp add: mult_commute [of _ "real (n - Suc 0)"])
apply (auto intro!: real_setsum_nat_ivl_bounded)
apply (case_tac "n", auto)
apply (drule less_add_one)
(*CLAIM_SIMP " (a * b * c = a * (c * (b::real))" mult_ac]*)
apply clarify 
apply (subgoal_tac "K ^ p * K ^ d * real (Suc (Suc (p + d))) =
                    K ^ p * (real (Suc (Suc (p + d))) * K ^ d)") 
apply (simp (no_asm_simp) add: power_add del: setsum_Suc)
apply (auto intro!: mult_mono simp del: setsum_Suc)
apply (auto intro!: power_mono simp add: power_abs simp del: setsum_Suc)
apply (rule_tac j = "real (Suc d) * (K ^ d)" in real_le_trans)
apply (subgoal_tac [2] "0 \<le> K")
apply (drule_tac [2] n = d in zero_le_power)
apply (auto simp del: setsum_Suc)
apply (rule setsum_abs [THEN real_le_trans])
apply (rule real_setsum_nat_ivl_bounded, auto dest!: less_add_one intro!: mult_mono simp add: power_add)
apply (auto intro!: power_mono zero_le_power simp add: power_abs, arith+)
done

lemma lemma_termdiff4: 
  "[| 0 < k;  
      (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>f h\<bar> \<le> K * \<bar>h\<bar>) |]  
   ==> f -- 0 --> 0"
apply (simp add: LIM_def, auto)
apply (subgoal_tac "0 \<le> K")
 prefer 2
 apply (drule_tac x = "k/2" in spec)
apply (simp add: ); 
 apply (subgoal_tac "0 \<le> K*k", simp add: zero_le_mult_iff) 
 apply (force intro: order_trans [of _ "\<bar>f (k / 2)\<bar> * 2"]) 
apply (drule real_le_imp_less_or_eq, auto)
apply (subgoal_tac "0 < (r * inverse K) / 2")
apply (drule_tac ?d1.0 = "(r * inverse K) / 2" and ?d2.0 = k in real_lbound_gt_zero)
apply (auto simp add: positive_imp_inverse_positive zero_less_mult_iff zero_less_divide_iff)
apply (rule_tac x = e in exI, auto)
apply (rule_tac y = "K * \<bar>x\<bar>" in order_le_less_trans)
apply (force ); 
apply (rule_tac y = "K * e" in order_less_trans)
apply (simp add: mult_less_cancel_left)
apply (rule_tac c = "inverse K" in mult_right_less_imp_less)
apply (auto simp add: mult_ac)
done

lemma lemma_termdiff5:
     "[| 0 < k;  
            summable f;  
            \<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k -->  
                    (\<forall>n. abs(g(h) (n::nat)) \<le> (f(n) * \<bar>h\<bar>)) |]  
         ==> (%h. suminf(g h)) -- 0 --> 0"
apply (drule summable_sums)
apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>suminf (g h)\<bar> \<le> suminf f * \<bar>h\<bar>")
apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
 prefer 2
 apply (simp add: summable_def) 
 apply (rule_tac x = "suminf f * \<bar>h\<bar>" in exI)
 apply (drule_tac c = "\<bar>h\<bar>" in sums_mult)
 apply (simp add: mult_ac) 
apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
 apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
  apply (rule_tac [2] x = 0 in exI, auto)
apply (rule_tac j = "\<Sum>n. \<bar>g h n\<bar>" in real_le_trans)
apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult])
done



text{* FIXME: Long proofs*}

lemma termdiffs_aux:
     "[|summable (\<lambda>n. diffs (diffs c) n * K ^ n); \<bar>x\<bar> < \<bar>K\<bar> |]
    ==> (\<lambda>h. \<Sum>n. c n *
                  (((x + h) ^ n - x ^ n) * inverse h -
                   real n * x ^ (n - Suc 0)))
       -- 0 --> 0"
apply (drule dense, safe)
apply (frule real_less_sum_gt_zero)
apply (drule_tac
         f = "%n. \<bar>c n\<bar> * real n * real (n - Suc 0) * (r ^ (n - 2))" 
     and g = "%h n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) 
                             - (real n * (x ^ (n - Suc 0))))" 
       in lemma_termdiff5)
apply (auto simp add: add_commute)
apply (subgoal_tac "summable (%n. \<bar>diffs (diffs c) n\<bar> * (r ^ n))")
apply (rule_tac [2] x = K in powser_insidea, auto)
apply (subgoal_tac [2] "\<bar>r\<bar> = r", auto)
apply (rule_tac [2] y1 = "\<bar>x\<bar>" in order_trans [THEN abs_eq], auto)
apply (simp add: diffs_def mult_assoc [symmetric])
apply (subgoal_tac
        "\<forall>n. real (Suc n) * real (Suc (Suc n)) * \<bar>c (Suc (Suc n))\<bar> * (r ^ n) 
              = diffs (diffs (%n. \<bar>c n\<bar>)) n * (r ^ n) ") 
apply auto
apply (drule diffs_equiv)
apply (drule sums_summable)
apply (simp_all add: diffs_def) 
apply (simp add: diffs_def mult_ac)
apply (subgoal_tac " (%n. real n * (real (Suc n) * (\<bar>c (Suc n)\<bar> * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * \<bar>c m\<bar> * inverse r) n * (r ^ n))")
apply auto
  prefer 2
  apply (rule ext)
  apply (simp add: diffs_def) 
  apply (case_tac "n", auto)
txt{*23*}
   apply (drule abs_ge_zero [THEN order_le_less_trans])
   apply (simp add: mult_ac) 
  apply (drule abs_ge_zero [THEN order_le_less_trans])
  apply (simp add: mult_ac) 
 apply (drule diffs_equiv)
 apply (drule sums_summable)
apply (subgoal_tac
          "summable
            (\<lambda>n. real n * (real (n - Suc 0) * \<bar>c n\<bar> * inverse r) *
                 r ^ (n - Suc 0)) =
           summable
            (\<lambda>n. real n * (\<bar>c n\<bar> * (real (n - Suc 0) * r ^ (n - 2))))")
apply simp 
apply (rule_tac f = summable in arg_cong, rule ext)
txt{*33*}
apply (case_tac "n", auto)
apply (case_tac "nat", auto)
apply (drule abs_ge_zero [THEN order_le_less_trans], auto) 
apply (drule abs_ge_zero [THEN order_le_less_trans])
apply (simp add: mult_assoc)
apply (rule mult_left_mono)
 prefer 2 apply arith 
apply (subst add_commute)
apply (simp (no_asm) add: mult_assoc [symmetric])
apply (rule lemma_termdiff3)
apply (auto intro: abs_triangle_ineq [THEN order_trans], arith)
done


lemma termdiffs: 
    "[| summable(%n. c(n) * (K ^ n));  
        summable(%n. (diffs c)(n) * (K ^ n));  
        summable(%n. (diffs(diffs c))(n) * (K ^ n));  
        \<bar>x\<bar> < \<bar>K\<bar> |]  
     ==> DERIV (%x. \<Sum>n. c(n) * (x ^ n))  x :>  
             (\<Sum>n. (diffs c)(n) * (x ^ n))"
apply (simp add: deriv_def)
apply (rule_tac g = "%h. \<Sum>n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h" in LIM_trans)
apply (simp add: LIM_def, safe)
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
apply (auto simp add: less_diff_eq)
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
apply (rule_tac y = 0 in order_le_less_trans, auto)
apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (\<Sum>n. (c n) * (x ^ n)) & (%n. (c n) * ((x + xa) ^ n)) sums (\<Sum>n. (c n) * ( (x + xa) ^ n))")
apply (auto intro!: summable_sums)
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
apply (auto simp add: add_commute)
apply (drule_tac x="(\<lambda>n. c n * (xa + x) ^ n)" in sums_diff, assumption) 
apply (drule_tac x = "(%n. c n * (xa + x) ^ n - c n * x ^ n) " and c = "inverse xa" in sums_mult)
apply (rule sums_unique)
apply (simp add: diff_def divide_inverse add_ac mult_ac)
apply (rule LIM_zero_cancel)
apply (rule_tac g = "%h. \<Sum>n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) - (real n * (x ^ (n - Suc 0))))" in LIM_trans)
 prefer 2 apply (blast intro: termdiffs_aux) 
apply (simp add: LIM_def, safe)
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
apply (auto simp add: less_diff_eq)
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
apply (rule_tac y = 0 in order_le_less_trans, auto)
apply (subgoal_tac "summable (%n. (diffs c) (n) * (x ^ n))")
apply (rule_tac [2] powser_inside, auto)
apply (drule_tac c = c and x = x in diffs_equiv)
apply (frule sums_unique, auto)
apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (\<Sum>n. (c n) * (x ^ n)) & (%n. (c n) * ((x + xa) ^ n)) sums (\<Sum>n. (c n) * ( (x + xa) ^ n))")
apply safe
apply (auto intro!: summable_sums)
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
apply (auto simp add: add_commute)
apply (frule_tac x = "(%n. c n * (xa + x) ^ n) " and y = "(%n. c n * x ^ n)" in sums_diff, assumption)
apply (simp add: suminf_diff [OF sums_summable sums_summable] 
               right_diff_distrib [symmetric])
apply (frule_tac x = "(%n. c n * ((xa + x) ^ n - x ^ n))" and c = "inverse xa" in sums_mult)
apply (simp add: sums_summable [THEN suminf_mult2])
apply (frule_tac x = "(%n. inverse xa * (c n * ((xa + x) ^ n - x ^ n))) " and y = "(%n. real n * c n * x ^ (n - Suc 0))" in sums_diff)
apply assumption
apply (simp add: suminf_diff [OF sums_summable sums_summable] add_ac mult_ac)
apply (rule_tac f = suminf in arg_cong)
apply (rule ext)
apply (simp add: diff_def right_distrib add_ac mult_ac)
done


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] del: mult_Suc)

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 simp del: mult_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 (auto intro!: ext 
         simp add: diffs_def divide_inverse simp del: mult_Suc)

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
         simp del: mult_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 (auto intro!: ext 
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
         simp del: mult_Suc)

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 = (%x. \<Sum>n. inverse (real (fact n)) * x ^ 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 (%u. \<Sum>n. inverse (real (fact n)) * u ^ n) x :> (\<Sum>n. diffs (%n. inverse (real (fact n))) n * x ^ n)")
apply (rule_tac [2] K = "1 + \<bar>x\<bar>" in termdiffs)
apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs, arith)
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, arith)
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, arith)
done


subsection{*Properties of the Exponential Function*}

lemma exp_zero [simp]: "exp 0 = 1"
proof -
  have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
        (\<Sum>n. inverse (real (fact n)) * 0 ^ n)"
    by (rule series_zero [rule_format, THEN sums_unique],
        case_tac "m", auto)
  thus ?thesis by (simp add:  exp_def) 
qed

lemma exp_ge_add_one_self [simp]: "0 \<le> x ==> (1 + x) \<le> exp(x)"
apply (drule real_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 ==> 1 < exp x"
apply (rule order_less_le_trans)
apply (rule_tac [2] exp_ge_add_one_self, 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_Id 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_Id) 
  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
qed

lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
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"
proof -
  have "exp (x + 0) * exp (- x) = exp 0" by (rule exp_add_mult_minus) 
  thus ?thesis by simp
qed

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])

lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
proof -
  have "exp x * exp y = exp x * (exp (x + y) * exp (- x))" by simp
  thus ?thesis by (simp (no_asm_simp) add: mult_ac)
qed

text{*Proof: because every exponential can be seen as a square.*}
lemma exp_ge_zero [simp]: "0 \<le> exp x"
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"
by (simp add: order_less_le)

lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
by (auto intro: positive_imp_inverse_positive)

lemma abs_exp_cancel [simp]: "\<bar>exp x\<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:
  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 < 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) < exp(y)) = (x < y)"
by (auto intro: exp_less_mono exp_less_cancel)

lemma exp_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
by (auto simp add: linorder_not_less [symmetric])

lemma exp_inj_iff [iff]: "(exp x = 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) = y"
apply (rule IVT)
apply (auto intro: DERIV_exp [THEN DERIV_isCont] simp add: le_diff_eq)
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
apply simp 
apply (rule exp_ge_add_one_self, simp)
done

lemma exp_total: "0 < y ==> \<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*}

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

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 (frule real_mult_order)
apply (auto simp add: exp_add exp_ln_iff [symmetric] simp del: exp_inj_iff exp_ln_iff)
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], auto)
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 exp_ln_iff [symmetric] del: exp_ln_iff)
  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 exp_ln_iff [symmetric] del: exp_ln_iff)
  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


subsection{*Basic Properties of the Trigonometric Functions*}

lemma sin_zero [simp]: "sin 0 = 0"
by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
         simp add: sin_def sums_def simp del: power_0_left)

lemma lemma_series_zero2:
 "(\<forall>m. n \<le> m --> f m = 0) --> f sums setsum f {0..<n}"
by (auto intro: series_zero)

lemma cos_zero [simp]: "cos 0 = 1"
apply (simp add: cos_def)
apply (rule sums_unique [symmetric])
apply (cut_tac n = 1 and f = "(%n. (if even n then (- 1) ^ (n div 2) / (real (fact n)) else 0) * 0 ^ n)" in lemma_series_zero2)
apply auto
done

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 real_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 real_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))"
apply (rule lemma_DERIV_subst)
apply (rule DERIV_cos_realpow2a, auto)
done

(* most useful *)
lemma DERIV_cos_cos_mult3 [simp]:
     "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
apply (rule lemma_DERIV_subst)
apply (rule DERIV_cos_cos_mult2, auto)
done

lemma DERIV_sin_circle_all: 
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
apply (simp only: diff_minus, safe)
apply (rule DERIV_add) 
apply (auto simp add: numeral_2_eq_2)
done

lemma DERIV_sin_circle_all_zero [simp]:
     "\<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 real_add_commute)
apply (simp (no_asm) del: realpow_Suc)
done

lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
apply (cut_tac x = x in sin_cos_squared_add2)
apply (auto simp add: numeral_2_eq_2)
done

lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])
apply (simp del: realpow_Suc)
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 del: realpow_Suc)
done

lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
by arith

lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
apply (auto simp add: linorder_not_less [symmetric])
apply (drule_tac n = "Suc 0" in power_gt1)
apply (auto simp del: realpow_Suc)
apply (drule_tac r1 = "cos x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
done

lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
apply (insert abs_sin_le_one [of x]) 
apply (simp add: abs_le_interval_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_interval_iff del: abs_sin_le_one) 
done

lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
apply (auto simp add: linorder_not_less [symmetric])
apply (drule_tac n = "Suc 0" in power_gt1)
apply (auto simp del: realpow_Suc)
apply (drule_tac r1 = "sin x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
done

lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
apply (insert abs_cos_le_one [of x]) 
apply (simp add: abs_le_interval_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_interval_iff del: abs_cos_le_one)
done

lemma DERIV_fun_pow: "DERIV g x :> m ==>  
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
apply (rule lemma_DERIV_subst)
apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
apply (rule DERIV_pow, auto)
done

lemma DERIV_fun_exp:
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
apply (rule lemma_DERIV_subst)
apply (rule_tac f = exp in DERIV_chain2)
apply (rule DERIV_exp, auto)
done

lemma DERIV_fun_sin:
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
apply (rule lemma_DERIV_subst)
apply (rule_tac f = sin in DERIV_chain2)
apply (rule DERIV_sin, auto)
done

lemma DERIV_fun_cos:
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
apply (rule lemma_DERIV_subst)
apply (rule_tac f = cos in DERIV_chain2)
apply (rule DERIV_cos, auto)
done

lemmas DERIV_intros = DERIV_Id DERIV_const DERIV_cos DERIV_cmult 
                    DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
                    DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
                    DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
                    DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos 

(* lemma *)