(*  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 = NthRoot + Fact + HSeries + EvenOdd + Lim:

(*????FOR RING_AND_FIELD*)

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 == suminf(%n. inverse(real (fact n)) * (x ^ n))"

    sin :: "real => real"

    "sin x == suminf(%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 == suminf(%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 (unfold 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 (unfold 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 (unfold 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 simp add: order_less_irrefl)

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 (unfold 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 (unfold 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 simp add: order_less_irrefl)

done

subsection{*Square Root*}

(*lcp: needed now because 2 is a binary numeral!*)

lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"

apply (simp (no_asm) del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 add: nat_numeral_0_eq_0 [symmetric])

done

lemma real_sqrt_zero [simp]: "sqrt 0 = 0"

by (unfold sqrt_def, auto)

lemma real_sqrt_one [simp]: "sqrt 1 = 1"

by (unfold sqrt_def, auto)

lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"

apply (unfold 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 add: real_sqrt_pow2_iff)

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 (unfold sqrt_def)

apply (subst numeral_2_eq_2)

apply (auto intro: real_root_pow_pos2 [THEN ssubst] real_root_pos2 [THEN ssubst] simp del: realpow_Suc)

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_mult_self_eq_zero_iff [simp]: "(r * r = 0) = (r = (0::real))"

by auto

lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"

apply (unfold sqrt_def root_def)

apply (subst numeral_2_eq_2)

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)

(*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 (unfold 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: real_sqrt_pow2_iff 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 abs_mult)

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 order_less_irrefl)

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: abs_mult 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 simp add: abs_mult inverse_mult_distrib)

apply (auto simp add: mult_assoc [symmetric] abs_eqI2 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 abs_inverse)

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 (unfold real_divide_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: power_abs [symmetric] abs_mult 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 (unfold real_divide_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: power_abs [symmetric] abs_mult 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"

apply (induct_tac "n", auto)

done

lemma lemma_STAR_cos [simp]: "0 < n -->  

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

apply (induct_tac "n", auto)

done

lemma lemma_STAR_cos1 [simp]: "0 < n -->  

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

apply (induct_tac "n", auto)

done

lemma lemma_STAR_cos2 [simp]: "sumr 1 n (%n. if even n  

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

                          0 ^ n  

                    else 0) = 0"

apply (induct_tac "n")

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

done

lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"

apply (unfold 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 (unfold 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 (unfold 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_tac "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:

     "sumr 0 (Suc n) (%p. (x ^ p) * y ^ ((Suc n) - p)) =  

      y * sumr 0 (Suc n) (%p. (x ^ p) * (y ^ (n - p)))"

apply (auto simp add: sumr_mult simp del: sumr_Suc)

apply (rule sumr_subst)

apply (intro strip)

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) * sumr 0 (Suc n) (%p. (x ^ p) * (y ^(n - p)))"

apply (induct_tac "n", simp)

apply (auto simp del: sumr_Suc)

apply (subst sumr_Suc)

apply (drule sym)

apply (auto simp add: lemma_realpow_diff_sumr right_distrib real_diff_def mult_ac simp del: sumr_Suc)

done

lemma lemma_realpow_rev_sumr: "sumr 0 (Suc n) (%p. (x ^ p) * (y ^ (n - p))) =  

      sumr 0 (Suc n) (%p. (x ^ (n - p)) * (y ^ p))"

apply (case_tac "x = y")

apply (auto simp add: mult_commute power_add [symmetric] simp del: sumr_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: sumr_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. abs(f(n)) * (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 * abs (z ^ n) * inverse (abs (x ^ n))" in summable_comparison_test)

apply (rule_tac x = 0 in exI, safe)

apply (subgoal_tac "0 < abs (x ^ n) ")

apply (rule_tac c="abs (x ^ n)" 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: abs_mult 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 "abs (z ^ n) * 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)

apply (rule_tac c="\<bar>x\<bar>" in mult_right_less_imp_less) 

apply (auto simp add: abs_mult [symmetric] mult_assoc)

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:

     "sumr 0 n (%n. (diffs c)(n) * (x ^ n)) =  

      sumr 0 n (%n. real n * c(n) * (x ^ (n - Suc 0))) +  

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

apply (induct_tac "n")

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

done

lemma lemma_diffs2: "sumr 0 n (%n. real n * c(n) * (x ^ (n - Suc 0))) =  

      sumr 0 n (%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  

         (suminf(%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:

     "sumr 0 m (%p. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  

        sumr 0 m (%p. (z ^ p) *  

        (((z + h) ^ (m - p)) - (z ^ (m - p))))"

apply (rule sumr_subst)

apply (auto simp add: right_distrib real_diff_def 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 * sumr 0 (n - Suc 0) (%p. (z ^ p) *  

               sumr 0 ((n - Suc 0) - p)  

                 (%q. ((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 sumr_Suc)

apply (auto simp add: lemma_realpow_rev_sumr simp del: sumr_Suc)

apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 

                sumdiff lemma_termdiff1 sumr_mult)

apply (auto intro!: sumr_subst simp add: real_diff_def real_add_assoc)

apply (simp add: diff_minus [symmetric] less_iff_Suc_add) 

apply (auto simp add: sumr_mult lemma_realpow_diff_sumr2 mult_ac simp

                 del: sumr_Suc realpow_Suc)

done

lemma lemma_termdiff3: "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; abs (z + h) \<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: abs_mult mult_commute) 

apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 

apply (rule sumr_rabs [THEN real_le_trans])

apply (simp add: mult_assoc [symmetric])

apply (simp add: mult_commute [of _ "real (n - Suc 0)"])

apply (auto intro!: sumr_bound2 simp add: abs_mult)

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: sumr_Suc)

apply (auto intro!: mult_mono simp del: sumr_Suc)

apply (auto intro!: power_mono simp add: power_abs simp del: sumr_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: sumr_Suc)

apply (rule sumr_rabs [THEN real_le_trans])

apply (rule sumr_bound2, auto dest!: less_add_one intro!: mult_mono simp add: abs_mult 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 --> abs(f h) \<le> K * \<bar>h\<bar>) |]  

   ==> f -- 0 --> 0"

apply (unfold LIM_def, auto)

apply (subgoal_tac "0 \<le> K")

apply (drule_tac [2] x = "k/2" in spec)

apply (frule_tac [2] real_less_half_sum)

apply (drule_tac [2] real_gt_half_sum)

apply (auto simp add: abs_eqI2)

apply (rule_tac [2] c = "k/2" in mult_right_le_imp_le)

apply (auto intro: abs_ge_zero [THEN real_le_trans])

apply (drule real_le_imp_less_or_eq, auto)

apply (subgoal_tac "0 < (r * inverse K) * inverse 2")

apply (rule_tac [2] real_mult_order)+

apply (drule_tac ?d1.0 = "r * inverse K * inverse 2" and ?d2.0 = k in real_lbound_gt_zero)

apply (auto simp add: positive_imp_inverse_positive zero_less_mult_iff)

apply (rule_tac [2] y="\<bar>f (k / 2)\<bar> * 2" in order_trans, auto)

apply (rule_tac x = e in exI, auto)

apply (rule_tac y = "K * \<bar>x\<bar>" in order_le_less_trans)

apply (rule_tac [2] y = "K * e" in order_less_trans)

apply (rule_tac [3] c = "inverse K" in mult_right_less_imp_less, force)

apply (simp add: mult_less_cancel_left)

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 --> abs (suminf (g h)) \<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 = "suminf (%n. abs (g h n))" 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. suminf

             (\<lambda>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. abs (c n) * 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_eqI1], 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) * (abs (c (Suc n)) * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * abs (c m) * 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)

apply (rule add_commute [THEN subst])

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. suminf (%n. c(n) * (x ^ n)))  x :>  

             suminf (%n. (diffs c)(n) * (x ^ n))"

apply (unfold deriv_def)

apply (rule_tac g = "%h. suminf (%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 (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%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 [symmetric])

apply (simp add: diff_def divide_inverse add_ac mult_ac)

apply (rule LIM_zero_cancel)

apply (rule_tac g = "%h. suminf (%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 (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%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 (subst minus_equation_iff, simp (no_asm))  

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)))"

apply (unfold diffs_def)

apply (rule ext)

apply (subst fact_Suc)

apply (subst real_of_nat_mult)

apply (subst inverse_mult_distrib)

apply (auto simp add: mult_assoc [symmetric])

done

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)"

apply (unfold diffs_def real_divide_def)

apply (rule ext)

apply (subst fact_Suc)

apply (subst real_of_nat_mult)

apply (subst even_nat_Suc)

apply (subst inverse_mult_distrib, auto)

done

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)"

apply (unfold diffs_def real_divide_def)

apply (subst fact_Suc)

apply (subst real_of_nat_mult)

apply (subst even_nat_Suc)

apply (subst inverse_mult_distrib, auto)

done

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))))"

apply (unfold diffs_def real_divide_def)

apply (rule ext)

apply (subst fact_Suc)

apply (subst real_of_nat_mult)

apply (auto simp add: mult_assoc odd_Suc_mult_two_ex)

done

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)))"

apply (unfold diffs_def real_divide_def)

apply (subst fact_Suc)

apply (subst real_of_nat_mult) 

apply (auto simp add: mult_assoc odd_Suc_mult_two_ex)

done

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

lemma lemma_sin_minus:

     "- sin x =

         suminf(%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. suminf (%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 (unfold exp_def)

apply (subst lemma_exp_ext)

apply (subgoal_tac "DERIV (%u. suminf (%n. inverse (real (fact n)) * u ^ n)) x :> suminf (%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. suminf(%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. suminf(%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 (unfold 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) =

        suminf (\<lambda>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 (unfold 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]: "abs(exp x) = exp x"

by (auto simp add: abs_eqI2)

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

apply (induct_tac "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 (unfold real_diff_def real_divide_def)

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)

qed

lemma exp_less_cancel: "exp x < exp y ==> x < y"

apply (rule ccontr) 

apply (simp add: linorder_not_less order_le_less) 

apply (auto dest: 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 (unfold real_divide_def)