Theory Transcendental (Isabelle2005: October 2005)

# Theory Transcendental

Up to index of Isabelle/HOL/HOL-Complex

theory Transcendental
imports NthRoot Fact HSeries EvenOdd Lim
begin

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

        sin :: "real => real"
        "sin x == ∑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 == ∑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 ≤ (x::real) & x ≤ 2 & cos x = 0)"

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

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

        arcos :: "real => real"
        "arcos y == (@x. 0 ≤ x & x ≤ 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 ≤ 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 ≤ 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 ≤ 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 ≤ x ==> 0 ≤ 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)² = x) = (0 ≤ 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(u2² + v2²))² = u2² + v2²"
    by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])

    lemma real_sqrt_pow2 [simp]: "0 ≤ x ==> (sqrt x)² = x"
    by (simp)

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

    lemma real_pow_sqrt_eq_sqrt_pow:
          "0 ≤ x ==> (sqrt x)² = sqrt(x²)"
    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 ≤ x ==> (sqrt x)² = sqrt(¦x¦ ^ 2)"
    by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])

    lemma real_sqrt_pow_abs: "0 ≤ x ==> (sqrt x)² = ¦x¦"
    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 ≤ x ==> 0 ≤ sqrt(x)"
    by (auto intro: real_sqrt_gt_zero simp add: order_le_less)

    lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 ≤ 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 --> 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: "[|r² = a; 0 ≤ 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 ≤ x; 0 ≤ 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≤x; 0≤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 ≤ sqrt (x² + y²)"
    by (auto intro!: real_sqrt_ge_zero)

    lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
         "0 ≤ sqrt ((x² + y²)*(xa² + ya²))"
    by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)

    lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
         "sqrt ((x² + y²) * (xa² + ya²)) ^ 2 = (x² + y²) * (xa² + ya²)"
    by (auto simp add: zero_le_mult_iff simp del: realpow_Suc)

    lemma real_sqrt_abs [simp]: "sqrt(x²) = ¦x¦"
    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) = ¦x¦"
    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)²"
    by simp

    lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x ≠ 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 ≤ 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 ≤ x ==> ((sqrt x = 0) = (x=0))"
    by (auto simp add: real_sqrt_eq_zero_cancel)

    lemma real_sqrt_sum_squares_ge1 [simp]: "x ≤ sqrt(x² + y²)"
    apply (subgoal_tac "x ≤ 0 | 0 ≤ 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 ≤ sqrt(z² + y²)"
    apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
    done

    lemma real_sqrt_ge_one: "1 ≤ x ==> 1 ≤ 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 = "¦x¦" 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 = "¦x¦" 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)) * ¦x¦ ^ 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)) * ¦x¦ ^ 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 [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]:
      "(∑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 ≤ 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:
         "(∑p=0..<Suc n. (x ^ p) * y ^ ((Suc n) - p)) =
          y * (∑p=0..<Suc n. (x ^ p) * (y ^ (n - p))::real)"
    by (auto simp add: setsum_mult lemma_realpow_diff mult_ac
      simp del: setsum_op_ivl_Suc cong: strong_setsum_cong)

    lemma lemma_realpow_diff_sumr2:
         "x ^ (Suc n) - y ^ (Suc n) =
          (x - y) * (∑p=0..<Suc n. (x ^ p) * (y ^(n - p))::real)"
    apply (induct "n", simp)
    apply (auto simp del: setsum_op_ivl_Suc)
    apply (subst setsum_op_ivl_Suc)
    apply (drule sym)
    apply (auto simp add: lemma_realpow_diff_sumr right_distrib diff_minus mult_ac simp del: setsum_op_ivl_Suc)
    done

    lemma lemma_realpow_rev_sumr:
         "(∑p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
          (∑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_op_ivl_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_op_ivl_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 "¦z¦ < ¦x¦"}.*}

    lemma powser_insidea:
         "[| summable (%n. f(n) * (x ^ n)); ¦z¦ < ¦x¦ |]
          ==> summable (%n. ¦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 * ¦z ^ n¦ * inverse (¦x ^ n¦)" in summable_comparison_test)
    apply (rule_tac x = 0 in exI, safe)
    apply (subgoal_tac "0 < ¦x ^ n¦ ")
    apply (rule_tac c="¦x ^ n¦" in mult_right_le_imp_le)
    apply (auto simp add: mult_assoc power_abs abs_mult)
     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 - (¦z¦ * inverse (¦x¦)))" in exI)
    apply (auto intro!: sums_mult simp add: mult_assoc)
    apply (subgoal_tac "¦z ^ n¦ * inverse (¦x¦ ^ n) = (¦z¦ * inverse (¦x¦)) ^ n")
    apply (auto simp add: power_abs [symmetric])
    apply (subgoal_tac "x ≠ 0")
    apply (subgoal_tac [3] "x ≠ 0")
    apply (auto simp del: abs_inverse
                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)); ¦z¦ < ¦x¦ |]
          ==> summable (%n. f(n) * (z ^ n))"
    apply (drule_tac z = "¦z¦" in powser_insidea)
    apply (auto intro: summable_rabs_cancel simp add: abs_mult 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:
         "(∑n=0..<n. (diffs c)(n) * (x ^ n)) =
          (∑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:
         "(∑n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) =
          (∑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
             (∑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="(λn. ∑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:
      "(∑p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
       (∑p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p)))::real)"
    by (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac
      cong: strong_setsum_cong)

    lemma less_add_one: "m < n ==> (∃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 ≠ 0 ==>
       (((z + h) ^ n) - (z ^ n)) * inverse h - real n * (z ^ (n - Suc 0)) =
       h * (∑p=0..< n - Suc 0. (z ^ p) *
           (∑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_op_ivl_Suc)
    apply (auto simp add: lemma_realpow_rev_sumr simp del: setsum_op_ivl_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_op_ivl_Suc realpow_Suc)
    done

    lemma lemma_termdiff3:
         "[| h ≠ 0; ¦z¦ ≤ K; ¦z + h¦ ≤ K |]
          ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))
              ≤ real n * real (n - Suc 0) * K ^ (n - 2) * ¦h¦"
    apply (subst lemma_termdiff2, assumption)
    apply (simp add: mult_commute abs_mult)
    apply (simp add: mult_commute [of _ "K ^ (n - 2)"])
    apply (rule setsum_abs [THEN real_le_trans])
    apply (simp add: mult_assoc [symmetric] abs_mult)
    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_op_ivl_Suc)
    apply (auto intro!: mult_mono simp del: setsum_op_ivl_Suc)
    apply (auto intro!: power_mono simp add: power_abs
               simp del: setsum_op_ivl_Suc)
    apply (rule_tac j = "real (Suc d) * (K ^ d)" in real_le_trans)
    apply (subgoal_tac [2] "0 ≤ K")
    apply (drule_tac [2] n = d in zero_le_power)
    apply (auto simp del: setsum_op_ivl_Suc)
    apply (rule setsum_abs [THEN real_le_trans])
    apply (rule real_setsum_nat_ivl_bounded)
    apply (auto dest!: less_add_one intro!: mult_mono simp add: power_add abs_mult)
    apply (auto intro!: power_mono zero_le_power simp add: power_abs, arith+)
    done

    lemma lemma_termdiff4:
      "[| 0 < k;
          (∀h. 0 < ¦h¦ & ¦h¦ < k --> ¦f h¦ ≤ K * ¦h¦) |]
       ==> f -- 0 --> 0"
    apply (simp add: LIM_def, auto)
    apply (subgoal_tac "0 ≤ K")
     prefer 2
     apply (drule_tac x = "k/2" in spec)
    apply (simp add: );
     apply (subgoal_tac "0 ≤ K*k", simp add: zero_le_mult_iff)
     apply (force intro: order_trans [of _ "¦f (k / 2)¦ * 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 * ¦x¦" 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;
                ∀h. 0 < ¦h¦ & ¦h¦ < k -->
                        (∀n. abs(g(h) (n::nat)) ≤ (f(n) * ¦h¦)) |]
             ==> (%h. suminf(g h)) -- 0 --> 0"
    apply (drule summable_sums)
    apply (subgoal_tac "∀h. 0 < ¦h¦ & ¦h¦ < k --> ¦suminf (g h)¦ ≤ suminf f * ¦h¦")
    apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
    apply (subgoal_tac "summable (%n. f n * ¦h¦) ")
     prefer 2
     apply (simp add: summable_def)
     apply (rule_tac x = "suminf f * ¦h¦" in exI)
     apply (drule_tac c = "¦h¦" 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 * ¦h¦" in summable_comparison_test)
      apply (rule_tac [2] x = 0 in exI, auto)
    apply (rule_tac j = "∑n. ¦g h n¦" in real_le_trans)
    apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult2])
    done

    text{* FIXME: Long proofs*}

    lemma termdiffs_aux:
         "[|summable (λn. diffs (diffs c) n * K ^ n); ¦x¦ < ¦K¦ |]
        ==> (λh. ∑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. ¦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. ¦diffs (diffs c) n¦ * (r ^ n))")
    apply (rule_tac [2] x = K in powser_insidea, auto)
    apply (subgoal_tac [2] "¦r¦ = r", auto)
    apply (rule_tac [2] y1 = "¦x¦" in order_trans [THEN abs_of_nonneg], auto)
    apply (simp add: diffs_def mult_assoc [symmetric])
    apply (subgoal_tac
            "∀n. real (Suc n) * real (Suc (Suc n)) * ¦c (Suc (Suc n))¦ * (r ^ n)
                  = diffs (diffs (%n. ¦c n¦)) n * (r ^ n) ")
    apply (auto simp add: abs_mult)
    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) * (¦c (Suc n)¦ * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * ¦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
                (λn. real n * (real (n - Suc 0) * ¦c n¦ * inverse r) *
                     r ^ (n - Suc 0)) =
               summable
                (λn. real n * (¦c n¦ * (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));
            ¦x¦ < ¦K¦ |]
         ==> DERIV (%x. ∑n. c(n) * (x ^ n))  x :>
                 (∑n. (diffs c)(n) * (x ^ n))"
    apply (simp add: deriv_def)
    apply (rule_tac g = "%h. ∑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 = "¦K¦ - ¦x¦" 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 (∑n. (c n) * (x ^ n)) & (%n. (c n) * ((x + xa) ^ n)) sums (∑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="(λn. c n * (xa + x) ^ n)" in sums_diff, assumption)
    apply (drule_tac f = "(%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. ∑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 = "¦K¦ - ¦x¦" 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 (∑n. (c n) * (x ^ n)) & (%n. (c n) * ((x + xa) ^ n)) sums (∑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 (subst suminf_diff)
    apply (rule summable_mult2)
    apply (erule sums_summable)
    apply (erule sums_summable)
    apply (simp add: ring_eq_simps)
    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 = (∑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. ∑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. ∑n. inverse (real (fact n)) * u ^ n) x :> (∑n. diffs (%n. inverse (real (fact n))) n * x ^ n)")
    apply (rule_tac [2] K = "1 + ¦x¦" in termdiffs)
    apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs, arith)
    done

    lemma lemma_sin_ext:
         "sin = (%x. ∑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. ∑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 + ¦x¦" 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 + ¦x¦" 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 "(∑n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
            (∑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_aux: "0 ≤ x ==> (1 + x) ≤ 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_aux, auto)
    done

    lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
    proof -
      have "DERIV (exp o (λx. 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 o 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 (λx. 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 "∀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 ≤ 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 ≠ 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]: "¦exp x¦ = 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) ≤ exp(y)) = (x ≤ 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 ≤ y ==> ∃x. 0 ≤ x & x ≤ 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) ≤ exp (y - 1)")
    apply simp
    apply (rule exp_ge_add_one_self_aux, simp)
    done

    lemma exp_total: "0 < y ==> ∃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 ≤ ln y) = (x ≤ 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 ≤ x ==> ln(1 + x) ≤ 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 ≤ 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_le_cancel_iff)
    qed

    lemma ln_ge_zero_imp_ge_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_ge_zero_iff [simp]: "0 < x ==> (0 ≤ ln x) = (1 ≤ 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:
     "(∀m. n ≤ 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)²) 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)²) 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)²) 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)²) 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)²) 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:
         "∀x. DERIV (%x. (sin x)² + (cos x)²) 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]:
         "∀x. DERIV (%x. (sin x)² + (cos x)²) x :> 0"
    by (cut_tac DERIV_sin_circle_all, auto)

    lemma sin_cos_squared_add [simp]: "((sin x)²) + ((cos x)²) = 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)²) + ((sin x)²) = 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)² = 1 - (cos x)²"
    apply (rule_tac a1 = "(cos x)²" in add_right_cancel [THEN iffD1])
    apply (simp del: realpow_Suc)
    done

    lemma cos_squared_eq: "(cos x)² = 1 - (sin x)²"
    apply (rule_tac a1 = "(sin x)²" in add_right_cancel [THEN iffD1])
    apply (simp del: realpow_Suc)
    done

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

    lemma abs_sin_le_one [simp]: "¦sin x¦ ≤ 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 ≤ 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 ≤ 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]: "¦cos x¦ ≤ 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 ≤ 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 ≤ 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 *)
    lemma lemma_DERIV_sin_cos_add:
         "∀x.
             DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
                   (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
    apply (safe, rule lemma_DERIV_subst)
    apply (best intro!: DERIV_intros intro: DERIV_chain2)
      --{*replaces the old @{text DERIV_tac}*}
    apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
    done

    lemma sin_cos_add [simp]:
         "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
          (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
    apply (cut_tac y = 0 and x = x and y7 = y
           in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
    apply (auto simp add: numeral_2_eq_2)
    done

    lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
    apply (cut_tac x = x and y = y in sin_cos_add)
    apply (auto dest!: real_sum_squares_cancel_a
                simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
    done

    lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
    apply (cut_tac x = x and y = y in sin_cos_add)
    apply (auto dest!: real_sum_squares_cancel_a
                simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
    done

    lemma lemma_DERIV_sin_cos_minus:
        "∀x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
    apply (safe, rule lemma_DERIV_subst)
    apply (best intro!: DERIV_intros intro: DERIV_chain2)
    apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
    done

    lemma sin_cos_minus [simp]:
        "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
    apply (cut_tac y = 0 and x = x
           in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
    apply (auto simp add: numeral_2_eq_2)
    done

    lemma sin_minus [simp]: "sin (-x) = -sin(x)"
    apply (cut_tac x = x in sin_cos_minus)
    apply (auto dest!: real_sum_squares_cancel_a
           simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_minus)
    done

    lemma cos_minus [simp]: "cos (-x) = cos(x)"
    apply (cut_tac x = x in sin_cos_minus)
    apply (auto dest!: real_sum_squares_cancel_a
                       simp add: numeral_2_eq_2 simp del: sin_cos_minus)
    done

    lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
    apply (simp add: diff_minus)
    apply (simp (no_asm) add: sin_add)
    done

    lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
    by (simp add: sin_diff mult_commute)

    lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
    apply (simp add: diff_minus)
    apply (simp (no_asm) add: cos_add)
    done

    lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
    by (simp add: cos_diff mult_commute)

    lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
    by (cut_tac x = x and y = x in sin_add, auto)

    lemma cos_double: "cos(2* x) = ((cos x)²) - ((sin x)²)"
    apply (cut_tac x = x and y = x in cos_add)
    apply (auto simp add: numeral_2_eq_2)
    done

    subsection{*The Constant Pi*}

    text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
       hence define pi.*}

    lemma sin_paired:
         "(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
          sums  sin x"
    proof -
      have "(λn. ∑k = n * 2..<n * 2 + 2.
                (if even k then 0
                 else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
                x ^ k)
            sums
            (∑n. (if even n then 0
                         else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
                        x ^ n)"
        by (rule sin_converges [THEN sums_summable, THEN sums_group], simp)
      thus ?thesis by (simp add: mult_ac sin_def)
    qed

    lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
    apply (subgoal_tac
           "(λn. ∑k = n * 2..<n * 2 + 2.
                  (- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1))
         sums (∑n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
     prefer 2
     apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp)
    apply (rotate_tac 2)
    apply (drule sin_paired [THEN sums_unique, THEN ssubst])
    apply (auto simp del: fact_Suc realpow_Suc)
    apply (frule sums_unique)
    apply (auto simp del: fact_Suc realpow_Suc)
    apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
    apply (auto simp del: fact_Suc realpow_Suc)
    apply (erule sums_summable)
    apply (case_tac "m=0")
    apply (simp (no_asm_simp))
    apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")
    apply (simp only: mult_less_cancel_left, simp)
    apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
    apply (subgoal_tac "x*x < 2*3", simp)
    apply (rule mult_strict_mono)
    apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
    apply (subst fact_Suc)
    apply (subst fact_Suc)
    apply (subst fact_Suc)
    apply (subst fact_Suc)
    apply (subst real_of_nat_mult)
    apply (subst real_of_nat_mult)
    apply (subst real_of_nat_mult)
    apply (subst real_of_nat_mult)
    apply (simp (no_asm) add: divide_inverse del: fact_Suc)
    apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
    apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right)
    apply (auto simp add: mult_assoc simp del: fact_Suc)
    apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right)
    apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
    apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)")
    apply (erule ssubst)+
    apply (auto simp del: fact_Suc)
    apply (subgoal_tac "0 < x ^ (4 * m) ")
     prefer 2 apply (simp only: zero_less_power)
    apply (simp (no_asm_simp) add: mult_less_cancel_left)
    apply (rule mult_strict_mono)
    apply (simp_all (no_asm_simp))
    done

    lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
    by (auto intro: sin_gt_zero)

    lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
    apply (cut_tac x = x in sin_gt_zero1)
    apply (auto simp add: cos_squared_eq cos_double)
    done

    lemma cos_paired:
         "(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
    proof -
      have "(λn. ∑k = n * 2..<n * 2 + 2.
                (if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
                x ^ k)
            sums
            (∑n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
                  x ^ n)"
        by (rule cos_converges [THEN sums_summable, THEN sums_group], simp)
      thus ?thesis by (simp add: mult_ac cos_def)
    qed

    declare zero_less_power [simp]

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

    lemma cos_two_less_zero: "cos (2) < 0"
    apply (cut_tac x = 2 in cos_paired)
    apply (drule sums_minus)
    apply (rule neg_less_iff_less [THEN iffD1])
    apply (frule sums_unique, auto)
    apply (rule_tac y =
     "∑n=0..< Suc(Suc(Suc 0)). - ((- 1) ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
           in order_less_trans)
    apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
    apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
    apply (rule sumr_pos_lt_pair)
    apply (erule sums_summable, safe)
    apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
                del: fact_Suc)
    apply (rule real_mult_inverse_cancel2)
    apply (rule real_of_nat_fact_gt_zero)+
    apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
    apply (subst fact_lemma)
    apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
    apply (simp only: real_of_nat_mult)
    apply (rule real_mult_less_mono, force)
      apply (rule_tac [3] real_of_nat_fact_gt_zero)
     prefer 2 apply force
    apply (rule real_of_nat_less_iff [THEN iffD2])
    apply (rule fact_less_mono, auto)
    done
    declare cos_two_less_zero [simp]
    declare cos_two_less_zero [THEN real_not_refl2, simp]
    declare cos_two_less_zero [THEN order_less_imp_le, simp]

    lemma cos_is_zero: "EX! x. 0 ≤ x & x ≤ 2 & cos x = 0"
    apply (subgoal_tac "∃x. 0 ≤ x & x ≤ 2 & cos x = 0")
    apply (rule_tac [2] IVT2)
    apply (auto intro: DERIV_isCont DERIV_cos)
    apply (cut_tac x = xa and y = y in linorder_less_linear)
    apply (rule ccontr)
    apply (subgoal_tac " (∀x. cos differentiable x) & (∀x. isCont cos x) ")
    apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
    apply (drule_tac f = cos in Rolle)
    apply (drule_tac [5] f = cos in Rolle)
    apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
    apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
    apply (assumption, rule_tac y=y in order_less_le_trans, simp_all)
    apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all)
    done

    lemma pi_half: "pi/2 = (@x. 0 ≤ x & x ≤ 2 & cos x = 0)"
    by (simp add: pi_def)

    lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
    apply (rule cos_is_zero [THEN ex1E])
    apply (auto intro!: someI2 simp add: pi_half)
    done

    lemma pi_half_gt_zero: "0 < pi / 2"
    apply (rule cos_is_zero [THEN ex1E])
    apply (auto simp add: pi_half)
    apply (rule someI2, blast, safe)
    apply (drule_tac y = xa in real_le_imp_less_or_eq)
    apply (safe, simp)
    done
    declare pi_half_gt_zero [simp]
    declare pi_half_gt_zero [THEN real_not_refl2, THEN not_sym, simp]
    declare pi_half_gt_zero [THEN order_less_imp_le, simp]

    lemma pi_half_less_two: "pi / 2 < 2"
    apply (rule cos_is_zero [THEN ex1E])
    apply (auto simp add: pi_half)
    apply (rule someI2, blast, safe)
    apply (drule_tac x = xa in order_le_imp_less_or_eq)
    apply (safe, simp)
    done
    declare pi_half_less_two [simp]
    declare pi_half_less_two [THEN real_not_refl2, simp]
    declare pi_half_less_two [THEN order_less_imp_le, simp]

    lemma pi_gt_zero [simp]: "0 < pi"
    apply (insert pi_half_gt_zero)
    apply (simp add: );
    done

    lemma pi_neq_zero [simp]: "pi ≠ 0"
    by (rule pi_gt_zero [THEN real_not_refl2, THEN not_sym])

    lemma pi_not_less_zero [simp]: "~ (pi < 0)"
    apply (insert pi_gt_zero)
    apply (blast elim: order_less_asym)
    done

    lemma pi_ge_zero [simp]: "0 ≤ pi"
    by (auto intro: order_less_imp_le)

    lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
    by auto

    lemma sin_pi_half [simp]: "sin(pi/2) = 1"
    apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
    apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
    apply (auto simp add: numeral_2_eq_2)
    done

    lemma cos_pi [simp]: "cos pi = -1"
    by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)

    lemma sin_pi [simp]: "sin pi = 0"
    by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)

    lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
    by (simp add: diff_minus cos_add)

    lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
    by (simp add: cos_add)
    declare minus_sin_cos_eq [symmetric, simp]

    lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
    by (simp add: diff_minus sin_add)
    declare sin_cos_eq [symmetric, simp] cos_sin_eq [symmetric, simp]

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

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

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

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

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

    lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
    apply (induct "n")
    apply (auto simp add: real_of_nat_Suc left_distrib)
    done

    lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
    proof -
      have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
      also have "... = -1 ^ n" by (rule cos_npi)
      finally show ?thesis .
    qed

    lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
    apply (induct "n")
    apply (auto simp add: real_of_nat_Suc left_distrib)
    done

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

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

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

    lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
    apply (rule sin_gt_zero, assumption)
    apply (rule order_less_trans, assumption)
    apply (rule pi_half_less_two)
    done

    lemma sin_less_zero:
      assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
    proof -
      have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2)
      thus ?thesis by simp
    qed

    lemma pi_less_4: "pi < 4"
    by (cut_tac pi_half_less_two, auto)

    lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
    apply (cut_tac pi_less_4)
    apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
    apply (force intro: DERIV_isCont DERIV_cos)
    apply (cut_tac cos_is_zero, safe)
    apply (rename_tac y z)
    apply (drule_tac x = y in spec)
    apply (drule_tac x = "pi/2" in spec, simp)
    done

    lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
    apply (rule_tac x = x and y = 0 in linorder_cases)
    apply (rule cos_minus [THEN subst])
    apply (rule cos_gt_zero)
    apply (auto intro: cos_gt_zero)
    done

    lemma cos_ge_zero: "[| -(pi/2) ≤ x; x ≤ pi/2 |] ==> 0 ≤ cos x"
    apply (auto simp add: order_le_less cos_gt_zero_pi)
    apply (subgoal_tac "x = pi/2", auto)
    done

    lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
    apply (subst sin_cos_eq)
    apply (rotate_tac 1)
    apply (drule real_sum_of_halves [THEN ssubst])
    apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
    done

    lemma sin_ge_zero: "[| 0 ≤ x; x ≤ pi |] ==> 0 ≤ sin x"
    by (auto simp add: order_le_less sin_gt_zero_pi)

    lemma cos_total: "[| -1 ≤ y; y ≤ 1 |] ==> EX! x. 0 ≤ x & x ≤ pi & (cos x = y)"
    apply (subgoal_tac "∃x. 0 ≤ x & x ≤ pi & cos x = y")
    apply (rule_tac [2] IVT2)
    apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
    apply (cut_tac x = xa and y = y in linorder_less_linear)
    apply (rule ccontr, auto)
    apply (drule_tac f = cos in Rolle)
    apply (drule_tac [5] f = cos in Rolle)
    apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
                dest!: DERIV_cos [THEN DERIV_unique]
                simp add: differentiable_def)
    apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
    done

    lemma sin_total:
         "[| -1 ≤ y; y ≤ 1 |] ==> EX! x. -(pi/2) ≤ x & x ≤ pi/2 & (sin x = y)"
    apply (rule ccontr)
    apply (subgoal_tac "∀x. (- (pi/2) ≤ x & x ≤ pi/2 & (sin x = y)) = (0 ≤ (x + pi/2) & (x + pi/2) ≤ pi & (cos (x + pi/2) = -y))")
    apply (erule swap)
    apply (simp del: minus_sin_cos_eq [symmetric])
    apply (cut_tac y="-y" in cos_total, simp) apply simp
    apply (erule ex1E)
    apply (rule_tac a = "x - (pi/2)" in ex1I)
    apply (simp (no_asm) add: real_add_assoc)
    apply (rotate_tac 3)
    apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)
    done

    lemma reals_Archimedean4:
         "[| 0 < y; 0 ≤ x |] ==> ∃n. real n * y ≤ x & x < real (Suc n) * y"
    apply (auto dest!: reals_Archimedean3)
    apply (drule_tac x = x in spec, clarify)
    apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
     prefer 2 apply (erule LeastI)
    apply (case_tac "LEAST m::nat. x < real m * y", simp)
    apply (subgoal_tac "~ x < real nat * y")
     prefer 2 apply (rule not_less_Least, simp, force)
    done

    (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
       now causes some unwanted re-arrangements of literals!   *)
    lemma cos_zero_lemma:
         "[| 0 ≤ x; cos x = 0 |] ==>
          ∃n::nat. ~even n & x = real n * (pi/2)"
    apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
    apply (subgoal_tac "0 ≤ x - real n * pi &
                        (x - real n * pi) ≤ pi & (cos (x - real n * pi) = 0) ")
    apply (auto simp add: compare_rls)
      prefer 3 apply (simp add: cos_diff)
     prefer 2 apply (simp add: real_of_nat_Suc left_distrib)
    apply (simp add: cos_diff)
    apply (subgoal_tac "EX! x. 0 ≤ x & x ≤ pi & cos x = 0")
    apply (rule_tac [2] cos_total, safe)
    apply (drule_tac x = "x - real n * pi" in spec)
    apply (drule_tac x = "pi/2" in spec)
    apply (simp add: cos_diff)
    apply (rule_tac x = "Suc (2 * n)" in exI)
    apply (simp add: real_of_nat_Suc left_distrib, auto)
    done

    lemma sin_zero_lemma:
         "[| 0 ≤ x; sin x = 0 |] ==>
          ∃n::nat. even n & x = real n * (pi/2)"
    apply (subgoal_tac "∃n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
     apply (clarify, rule_tac x = "n - 1" in exI)
     apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
    apply (rule cos_zero_lemma)
    apply (simp_all add: add_increasing)
    done

    lemma cos_zero_iff:
         "(cos x = 0) =
          ((∃n::nat. ~even n & (x = real n * (pi/2))) |
           (∃n::nat. ~even n & (x = -(real n * (pi/2)))))"
    apply (rule iffI)
    apply (cut_tac linorder_linear [of 0 x], safe)
    apply (drule cos_zero_lemma, assumption+)
    apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
    apply (force simp add: minus_equation_iff [of x])
    apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
    apply (auto simp add: cos_add)
    done

    (* ditto: but to a lesser extent *)
    lemma sin_zero_iff:
         "(sin x = 0) =
          ((∃n::nat. even n & (x = real n * (pi/2))) |
           (∃n::nat. even n & (x = -(real n * (pi/2)))))"
    apply (rule iffI)
    apply (cut_tac linorder_linear [of 0 x], safe)
    apply (drule sin_zero_lemma, assumption+)
    apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
    apply (force simp add: minus_equation_iff [of x])
    apply (auto simp add: even_mult_two_ex)
    done

    subsection{*Tangent*}

    lemma tan_zero [simp]: "tan 0 = 0"
    by (simp add: tan_def)

    lemma tan_pi [simp]: "tan pi = 0"
    by (simp add: tan_def)

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

    lemma tan_minus [simp]: "tan (-x) = - tan x"
    by (simp add: tan_def minus_mult_left)

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

    lemma lemma_tan_add1:
          "[| cos x ≠ 0; cos y ≠ 0 |]
            ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
    apply (simp add: tan_def divide_inverse)
    apply (auto simp del: inverse_mult_distrib
                simp add: inverse_mult_distrib [symmetric] mult_ac)
    apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
    apply (auto simp del: inverse_mult_distrib
                simp add: mult_assoc left_diff_distrib cos_add)
    done

    lemma add_tan_eq:
          "[| cos x ≠ 0; cos y ≠ 0 |]
           ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
    apply (simp add: tan_def)
    apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
    apply (auto simp add: mult_assoc left_distrib)
    apply (simp add: sin_add)
    done

    lemma tan_add:
         "[| cos x ≠ 0; cos y ≠ 0; cos (x + y) ≠ 0 |]
          ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
    apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
    apply (simp add: tan_def)
    done

    lemma tan_double:
         "[| cos x ≠ 0; cos (2 * x) ≠ 0 |]
          ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
    apply (insert tan_add [of x x])
    apply (simp add: mult_2 [symmetric])
    apply (auto simp add: numeral_2_eq_2)
    done

    lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
    by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)

    lemma tan_less_zero:
      assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
    proof -
      have "0 < tan (- x)" using prems by (simp only: tan_gt_zero)
      thus ?thesis by simp
    qed

    lemma lemma_DERIV_tan:
         "cos x ≠ 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)²)"
    apply (rule lemma_DERIV_subst)
    apply (best intro!: DERIV_intros intro: DERIV_chain2)
    apply (auto simp add: divide_inverse numeral_2_eq_2)
    done

    lemma DERIV_tan [simp]: "cos x ≠ 0 ==> DERIV tan x :> inverse((cos x)²)"
    by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])

    lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
    apply (subgoal_tac "(λx. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
    apply (simp add: divide_inverse [symmetric])
    apply (rule LIM_mult2)
    apply (rule_tac [2] inverse_1 [THEN subst])
    apply (rule_tac [2] LIM_inverse)
    apply (simp_all add: divide_inverse [symmetric])
    apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric])
    apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
    done

    lemma lemma_tan_total: "0 < y ==> ∃x. 0 < x & x < pi/2 & y < tan x"
    apply (cut_tac LIM_cos_div_sin)
    apply (simp only: LIM_def)
    apply (drule_tac x = "inverse y" in spec, safe, force)
    apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
    apply (rule_tac x = "(pi/2) - e" in exI)
    apply (simp (no_asm_simp))
    apply (drule_tac x = "(pi/2) - e" in spec)
    apply (auto simp add: tan_def)
    apply (rule inverse_less_iff_less [THEN iffD1])
    apply (auto simp add: divide_inverse)
    apply (rule real_mult_order)
    apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
    apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
    done

    lemma tan_total_pos: "0 ≤ y ==> ∃x. 0 ≤ x & x < pi/2 & tan x = y"
    apply (frule real_le_imp_less_or_eq, safe)
     prefer 2 apply force
    apply (drule lemma_tan_total, safe)
    apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
    apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
    apply (drule_tac y = xa in order_le_imp_less_or_eq)
    apply (auto dest: cos_gt_zero)
    done

    lemma lemma_tan_total1: "∃x. -(pi/2) < x & x < (pi/2) & tan x = y"
    apply (cut_tac linorder_linear [of 0 y], safe)
    apply (drule tan_total_pos)
    apply (cut_tac [2] y="-y" in tan_total_pos, safe)
    apply (rule_tac [3] x = "-x" in exI)
    apply (auto intro!: exI)
    done

    lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
    apply (cut_tac y = y in lemma_tan_total1, auto)
    apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
    apply (subgoal_tac [2] "∃z. y < z & z < xa & DERIV tan z :> 0")
    apply (subgoal_tac "∃z. xa < z & z < y & DERIV tan z :> 0")
    apply (rule_tac [4] Rolle)
    apply (rule_tac [2] Rolle)
    apply (auto intro!: DERIV_tan DERIV_isCont exI
                simp add: differentiable_def)
    txt{*Now, simulate TRYALL*}
    apply (rule_tac [!] DERIV_tan asm_rl)
    apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
                simp add: cos_gt_zero_pi [THEN real_not_refl2, THEN not_sym])
    done

    lemma arcsin_pi:
         "[| -1 ≤ y; y ≤ 1 |]
          ==> -(pi/2) ≤ arcsin y & arcsin y ≤ pi & sin(arcsin y) = y"
    apply (drule sin_total, assumption)
    apply (erule ex1E)
    apply (simp add: arcsin_def)
    apply (rule someI2, blast)
    apply (force intro: order_trans)
    done

    lemma arcsin:
         "[| -1 ≤ y; y ≤ 1 |]
          ==> -(pi/2) ≤ arcsin y &
               arcsin y ≤ pi/2 & sin(arcsin y) = y"
    apply (unfold arcsin_def)
    apply (drule sin_total, assumption)
    apply (fast intro: someI2)
    done

    lemma sin_arcsin [simp]: "[| -1 ≤ y; y ≤ 1 |] ==> sin(arcsin y) = y"
    by (blast dest: arcsin)

    lemma arcsin_bounded:
         "[| -1 ≤ y; y ≤ 1 |] ==> -(pi/2) ≤ arcsin y & arcsin y ≤ pi/2"
    by (blast dest: arcsin)

    lemma arcsin_lbound: "[| -1 ≤ y; y ≤ 1 |] ==> -(pi/2) ≤ arcsin y"
    by (blast dest: arcsin)

    lemma arcsin_ubound: "[| -1 ≤ y; y ≤ 1 |] ==> arcsin y ≤ pi/2"
    by (blast dest: arcsin)

    lemma arcsin_lt_bounded:
         "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
    apply (frule order_less_imp_le)
    apply (frule_tac y = y in order_less_imp_le)
    apply (frule arcsin_bounded)
    apply (safe, simp)
    apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
    apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
    apply (drule_tac [!] f = sin in arg_cong, auto)
    done

    lemma arcsin_sin: "[|-(pi/2) ≤ x; x ≤ pi/2 |] ==> arcsin(sin x) = x"
    apply (unfold arcsin_def)
    apply (rule some1_equality)
    apply (rule sin_total, auto)
    done

    lemma arcos:
         "[| -1 ≤ y; y ≤ 1 |]
          ==> 0 ≤ arcos y & arcos y ≤ pi & cos(arcos y) = y"
    apply (simp add: arcos_def)
    apply (drule cos_total, assumption)
    apply (fast intro: someI2)
    done

    lemma cos_arcos [simp]: "[| -1 ≤ y; y ≤ 1 |] ==> cos(arcos y) = y"
    by (blast dest: arcos)

    lemma arcos_bounded: "[| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arcos y & arcos y ≤ pi"
    by (blast dest: arcos)

    lemma arcos_lbound: "[| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arcos y"
    by (blast dest: arcos)

    lemma arcos_ubound: "[| -1 ≤ y; y ≤ 1 |] ==> arcos y ≤ pi"
    by (blast dest: arcos)

    lemma arcos_lt_bounded:
         "[| -1 < y; y < 1 |]
          ==> 0 < arcos y & arcos y < pi"
    apply (frule order_less_imp_le)
    apply (frule_tac y = y in order_less_imp_le)
    apply (frule arcos_bounded, auto)
    apply (drule_tac y = "arcos y" in order_le_imp_less_or_eq)
    apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
    apply (drule_tac [!] f = cos in arg_cong, auto)
    done

    lemma arcos_cos: "[|0 ≤ x; x ≤ pi |] ==> arcos(cos x) = x"
    apply (simp add: arcos_def)
    apply (auto intro!: some1_equality cos_total)
    done

    lemma arcos_cos2: "[|x ≤ 0; -pi ≤ x |] ==> arcos(cos x) = -x"
    apply (simp add: arcos_def)
    apply (auto intro!: some1_equality cos_total)
    done

    lemma arctan [simp]:
         "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
    apply (cut_tac y = y in tan_total)
    apply (simp add: arctan_def)
    apply (fast intro: someI2)
    done

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

    lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
    by (auto simp only: arctan)

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

    lemma arctan_ubound: "arctan y < pi/2"
    by (auto simp only: arctan)

    lemma arctan_tan:
          "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
    apply (unfold arctan_def)
    apply (rule some1_equality)
    apply (rule tan_total, auto)
    done

    lemma arctan_zero_zero [simp]: "arctan 0 = 0"
    by (insert arctan_tan [of 0], simp)

    lemma cos_arctan_not_zero [simp]: "cos(arctan x) ≠ 0"
    apply (auto simp add: cos_zero_iff)
    apply (case_tac "n")
    apply (case_tac [3] "n")
    apply (cut_tac [2] y = x in arctan_ubound)
    apply (cut_tac [4] y = x in arctan_lbound)
    apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
    done

    lemma tan_sec: "cos x ≠ 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
    apply (rule power_inverse [THEN subst])
    apply (rule_tac c1 = "(cos x)²" in real_mult_right_cancel [THEN iffD1])
    apply (auto dest: realpow_not_zero
            simp add: power_mult_distrib left_distrib realpow_divide tan_def
                      mult_assoc power_inverse [symmetric]
            simp del: realpow_Suc)
    done

    text{*NEEDED??*}
    lemma [simp]:
         "sin (x + 1 / 2 * real (Suc m) * pi) =
          cos (x + 1 / 2 * real  (m) * pi)"
    by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)

    text{*NEEDED??*}
    lemma [simp]:
         "sin (x + real (Suc m) * pi / 2) =
          cos (x + real (m) * pi / 2)"
    by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)

    lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
    apply (rule lemma_DERIV_subst)
    apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
    apply (best intro!: DERIV_intros intro: DERIV_chain2)+
    apply (simp (no_asm))
    done

    lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
    proof -
      have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
        by (auto simp add: right_distrib sin_add left_distrib mult_ac)
      thus ?thesis
        by (simp add: real_of_nat_Suc left_distrib add_divide_distrib
                      mult_commute [of pi])
    qed

    lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
    by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)

    lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
    apply (subgoal_tac "3/2 = (1+1 / 2::real)")
    apply (simp only: left_distrib)
    apply (auto simp add: cos_add mult_ac)
    done

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

    lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
    apply (subgoal_tac "3/2 = (1+1 / 2::real)")
    apply (simp only: left_distrib)
    apply (auto simp add: sin_add mult_ac)
    done

    (*NEEDED??*)
    lemma [simp]:
         "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
    apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
    done

    (*NEEDED??*)
    lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
    by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)

    lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
    by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)

    lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
    apply (rule lemma_DERIV_subst)
    apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
    apply (best intro!: DERIV_intros intro: DERIV_chain2)+
    apply (simp (no_asm))
    done

    lemma isCont_cos [simp]: "isCont cos x"
    by (rule DERIV_cos [THEN DERIV_isCont])

    lemma isCont_sin [simp]: "isCont sin x"
    by (rule DERIV_sin [THEN DERIV_isCont])

    lemma isCont_exp [simp]: "isCont exp x"
    by (rule DERIV_exp [THEN DERIV_isCont])

    lemma sin_zero_abs_cos_one: "sin x = 0 ==> ¦cos x¦ = 1"
    by (auto simp add: sin_zero_iff even_mult_two_ex)

    lemma exp_eq_one_iff [simp]: "(exp x = 1) = (x = 0)"
    apply auto
    apply (drule_tac f = ln in arg_cong, auto)
    done

    lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
    by (cut_tac x = x in sin_cos_squared_add3, auto)

    lemma real_root_less_mono:
         "[| 0 ≤ x; x < y |] ==> root(Suc n) x < root(Suc n) y"
    apply (frule order_le_less_trans, assumption)
    apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst])
    apply (rotate_tac 1, assumption)
    apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst])
    apply (rotate_tac 3, assumption)
    apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le)
    apply (frule_tac n = n in real_root_pos_pos_le)
    apply (frule_tac n = n in real_root_pos_pos)
    apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing)
    apply (assumption, assumption)
    apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq)
    apply auto
    apply (drule_tac f = "%x. x ^ (Suc n)" in arg_cong)
    apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc)
    done

    lemma real_root_le_mono:
         "[| 0 ≤ x; x ≤ y |] ==> root(Suc n) x ≤ root(Suc n) y"
    apply (drule_tac y = y in order_le_imp_less_or_eq)
    apply (auto dest: real_root_less_mono intro: order_less_imp_le)
    done

    lemma real_root_less_iff [simp]:
         "[| 0 ≤ x; 0 ≤ y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)"
    apply (auto intro: real_root_less_mono)
    apply (rule ccontr, drule linorder_not_less [THEN iffD1])
    apply (drule_tac x = y and n = n in real_root_le_mono, auto)
    done

    lemma real_root_le_iff [simp]:
         "[| 0 ≤ x; 0 ≤ y |] ==> (root(Suc n) x ≤ root(Suc n) y) = (x ≤ y)"
    apply (auto intro: real_root_le_mono)
    apply (simp (no_asm) add: linorder_not_less [symmetric])
    apply auto
    apply (drule_tac x = y and n = n in real_root_less_mono, auto)
    done

    lemma real_root_eq_iff [simp]:
         "[| 0 ≤ x; 0 ≤ y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)"
    apply (auto intro!: order_antisym)
    apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1])
    apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto)
    done

    lemma real_root_pos_unique:
         "[| 0 ≤ x; 0 ≤ y; y ^ (Suc n) = x |] ==> root (Suc n) x = y"
    by (auto dest: real_root_pos2 simp del: realpow_Suc)

    lemma real_root_mult:
         "[| 0 ≤ x; 0 ≤ y |]
          ==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y"
    apply (rule real_root_pos_unique)
    apply (auto intro!: real_root_pos_pos_le
                simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2
                simp del: realpow_Suc)
    done

    lemma real_root_inverse:
         "0 ≤ x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))"
    apply (rule real_root_pos_unique)
    apply (auto intro: real_root_pos_pos_le
                simp add: power_inverse [symmetric] real_root_pow_pos2
                simp del: realpow_Suc)
    done

    lemma real_root_divide:
         "[| 0 ≤ x; 0 ≤ y |]
          ==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)"
    apply (simp add: divide_inverse)
    apply (auto simp add: real_root_mult real_root_inverse)
    done

    lemma real_sqrt_less_mono: "[| 0 ≤ x; x < y |] ==> sqrt(x) < sqrt(y)"
    by (simp add: sqrt_def)

    lemma real_sqrt_le_mono: "[| 0 ≤ x; x ≤ y |] ==> sqrt(x) ≤ sqrt(y)"
    by (simp add: sqrt_def)

    lemma real_sqrt_less_iff [simp]:
         "[| 0 ≤ x; 0 ≤ y |] ==> (sqrt(x) < sqrt(y)) = (x < y)"
    by (simp add: sqrt_def)

    lemma real_sqrt_le_iff [simp]:
         "[| 0 ≤ x; 0 ≤ y |] ==> (sqrt(x) ≤ sqrt(y)) = (x ≤ y)"
    by (simp add: sqrt_def)

    lemma real_sqrt_eq_iff [simp]:
         "[| 0 ≤ x; 0 ≤ y |] ==> (sqrt(x) = sqrt(y)) = (x = y)"
    by (simp add: sqrt_def)

    lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x² + y²) < 1) = (x² + y² < 1)"
    apply (rule real_sqrt_one [THEN subst], safe)
    apply (rule_tac [2] real_sqrt_less_mono)
    apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto)
    done

    lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x² + y²) = 1) = (x² + y² = 1)"
    apply (rule real_sqrt_one [THEN subst], safe)
    apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto)
    done

    lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
    apply (simp add: divide_inverse)
    apply (case_tac "r=0")
    apply (auto simp add: mult_ac)
    done

    subsection{*Theorems About Sqrt, Transcendental Functions for Complex*}

    lemma le_real_sqrt_sumsq [simp]: "x ≤ sqrt (x * x + y * y)"
    proof (rule order_trans)
      show "x ≤ sqrt(x*x)" by (simp add: abs_if)
      show "sqrt (x * x) ≤ sqrt (x * x + y * y)"
        by (rule real_sqrt_le_mono, auto)
    qed

    lemma minus_le_real_sqrt_sumsq [simp]: "-x ≤ sqrt (x * x + y * y)"
    proof (rule order_trans)
      show "-x ≤ sqrt(x*x)" by (simp add: abs_if)
      show "sqrt (x * x) ≤ sqrt (x * x + y * y)"
        by (rule real_sqrt_le_mono, auto)
    qed

    lemma lemma_real_divide_sqrt_ge_minus_one:
         "0 < x ==> -1 ≤ x/(sqrt (x * x + y * y))"
    by (simp add: divide_const_simps linorder_not_le [symmetric])

    lemma real_sqrt_sum_squares_gt_zero1: "x < 0 ==> 0 < sqrt (x * x + y * y)"
    apply (rule real_sqrt_gt_zero)
    apply (subgoal_tac "0 < x*x & 0 ≤ y*y", arith)
    apply (auto simp add: zero_less_mult_iff)
    done

    lemma real_sqrt_sum_squares_gt_zero2: "0 < x ==> 0 < sqrt (x * x + y * y)"
    apply (rule real_sqrt_gt_zero)
    apply (subgoal_tac "0 < x*x & 0 ≤ y*y", arith)
    apply (auto simp add: zero_less_mult_iff)
    done

    lemma real_sqrt_sum_squares_gt_zero3: "x ≠ 0 ==> 0 < sqrt(x² + y²)"
    apply (cut_tac x = x and y = 0 in linorder_less_linear)
    apply (auto intro: real_sqrt_sum_squares_gt_zero2 real_sqrt_sum_squares_gt_zero1 simp add: numeral_2_eq_2)
    done

    lemma real_sqrt_sum_squares_gt_zero3a: "y ≠ 0 ==> 0 < sqrt(x² + y²)"
    apply (drule_tac y = x in real_sqrt_sum_squares_gt_zero3)
    apply (auto simp add: real_add_commute)
    done

    lemma real_sqrt_sum_squares_eq_cancel: "sqrt(x² + y²) = x ==> y = 0"
    by (drule_tac f = "%x. x²" in arg_cong, auto)

    lemma real_sqrt_sum_squares_eq_cancel2: "sqrt(x² + y²) = y ==> x = 0"
    apply (rule_tac x = y in real_sqrt_sum_squares_eq_cancel)
    apply (simp add: real_add_commute)
    done

    lemma lemma_real_divide_sqrt_le_one: "x < 0 ==> x/(sqrt (x * x + y * y)) ≤ 1"
    by (insert lemma_real_divide_sqrt_ge_minus_one [of "-x" y], simp)

    lemma lemma_real_divide_sqrt_ge_minus_one2:
         "x < 0 ==> -1 ≤ x/(sqrt (x * x + y * y))"
    apply (simp add: divide_const_simps)
    apply (insert minus_le_real_sqrt_sumsq [of x y], arith)
    done

    lemma lemma_real_divide_sqrt_le_one2: "0 < x ==> x/(sqrt (x * x + y * y)) ≤ 1"
    by (cut_tac x = "-x" and y = y in lemma_real_divide_sqrt_ge_minus_one2, auto)

    lemma minus_sqrt_le: "- sqrt (x * x + y * y) ≤ x"
    by (insert minus_le_real_sqrt_sumsq [of x y], arith)

    lemma minus_sqrt_le2: "- sqrt (x * x + y * y) ≤ y"
    by (subst add_commute, simp add: minus_sqrt_le)

    lemma not_neg_sqrt_sumsq: "~ sqrt (x * x + y * y) < 0"
    by (simp add: linorder_not_less)

    lemma cos_x_y_ge_minus_one: "-1 ≤ x / sqrt (x * x + y * y)"
    by (simp add: minus_sqrt_le not_neg_sqrt_sumsq divide_const_simps)

    lemma cos_x_y_ge_minus_one1a [simp]: "-1 ≤ y / sqrt (x * x + y * y)"
    by (subst add_commute, simp add: cos_x_y_ge_minus_one)

    lemma cos_x_y_le_one [simp]: "x / sqrt (x * x + y * y) ≤ 1"
    apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
    apply (rule lemma_real_divide_sqrt_le_one)
    apply (rule_tac [3] lemma_real_divide_sqrt_le_one2, auto)
    done

    lemma cos_x_y_le_one2 [simp]: "y / sqrt (x * x + y * y) ≤ 1"
    apply (cut_tac x = y and y = x in cos_x_y_le_one)
    apply (simp add: real_add_commute)
    done

    declare cos_arcos [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp]
    declare arcos_bounded [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp]

    declare cos_arcos [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp]
    declare arcos_bounded [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp]

    lemma cos_abs_x_y_ge_minus_one [simp]:
         "-1 ≤ ¦x¦ / sqrt (x * x + y * y)"
    by (auto simp add: divide_const_simps abs_if linorder_not_le [symmetric])

    lemma cos_abs_x_y_le_one [simp]: "¦x¦ / sqrt (x * x + y * y) ≤ 1"
    apply (insert minus_le_real_sqrt_sumsq [of x y] le_real_sqrt_sumsq [of x y])
    apply (auto simp add: divide_const_simps abs_if linorder_neq_iff)
    done

    declare cos_arcos [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp]
    declare arcos_bounded [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp]

    lemma minus_pi_less_zero: "-pi < 0"
    by simp

    declare minus_pi_less_zero [simp]
    declare minus_pi_less_zero [THEN order_less_imp_le, simp]

    lemma arcos_ge_minus_pi: "[| -1 ≤ y; y ≤ 1 |] ==> -pi ≤ arcos y"
    apply (rule real_le_trans)
    apply (rule_tac [2] arcos_lbound, auto)
    done

    declare arcos_ge_minus_pi [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp]

    (* How tedious! *)
    lemma lemma_divide_rearrange:
         "[| x + (y::real) ≠ 0; 1 - z = x/(x + y) |] ==> z = y/(x + y)"
    apply (rule_tac c1 = "x + y" in real_mult_right_cancel [THEN iffD1])
    apply (frule_tac [2] c1 = "x + y" in real_mult_right_cancel [THEN iffD2])
    prefer 2 apply assumption
    apply (rotate_tac [2] 2)
    apply (drule_tac [2] mult_assoc [THEN subst])
    apply (rotate_tac [2] 2)
    apply (frule_tac [2] left_inverse [THEN subst])
    prefer 2 apply assumption
    apply (erule_tac [2] V = "(1 - z) * (x + y) = x / (x + y) * (x + y)" in thin_rl)
    apply (erule_tac [2] V = "1 - z = x / (x + y)" in thin_rl)
    apply (auto simp add: mult_assoc)
    apply (auto simp add: right_distrib left_diff_distrib)
    done

    lemma lemma_cos_sin_eq:
         "[| 0 < x * x + y * y;
             1 - (sin xa)² = (x / sqrt (x * x + y * y)) ^ 2 |]
          ==> (sin xa)² = (y / sqrt (x * x + y * y)) ^ 2"
    by (auto intro: lemma_divide_rearrange
             simp add: realpow_divide power2_eq_square [symmetric])

    lemma lemma_sin_cos_eq:
         "[| 0 < x * x + y * y;
             1 - (cos xa)² = (y / sqrt (x * x + y * y)) ^ 2 |]
          ==> (cos xa)² = (x / sqrt (x * x + y * y)) ^ 2"
    apply (auto simp add: realpow_divide power2_eq_square [symmetric])
    apply (subst add_commute)
    apply (rule lemma_divide_rearrange, simp add: real_add_eq_0_iff)
    apply (simp add: add_commute)
    done

    lemma sin_x_y_disj:
         "[| x ≠ 0;
             cos xa = x / sqrt (x * x + y * y)
          |] ==>  sin xa = y / sqrt (x * x + y * y) |
                  sin xa = - y / sqrt (x * x + y * y)"
    apply (drule_tac f = "%x. x²" in arg_cong)
    apply (frule_tac y = y in real_sum_square_gt_zero)
    apply (simp add: cos_squared_eq)
    apply (subgoal_tac "(sin xa)² = (y / sqrt (x * x + y * y)) ^ 2")
    apply (rule_tac [2] lemma_cos_sin_eq)
    apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
    done

    lemma lemma_cos_not_eq_zero: "x ≠ 0 ==> x / sqrt (x * x + y * y) ≠ 0"
    apply (simp add: divide_inverse)
    apply (frule_tac y3 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym, THEN nonzero_imp_inverse_nonzero])
    apply (auto simp add: power2_eq_square)
    done

    lemma cos_x_y_disj:
         "[| x ≠ 0;
             sin xa = y / sqrt (x * x + y * y)
          |] ==>  cos xa = x / sqrt (x * x + y * y) |
                  cos xa = - x / sqrt (x * x + y * y)"
    apply (drule_tac f = "%x. x²" in arg_cong)
    apply (frule_tac y = y in real_sum_square_gt_zero)
    apply (simp add: sin_squared_eq del: realpow_Suc)
    apply (subgoal_tac "(cos xa)² = (x / sqrt (x * x + y * y)) ^ 2")
    apply (rule_tac [2] lemma_sin_cos_eq)
    apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
    done

    lemma real_sqrt_divide_less_zero: "0 < y ==> - y / sqrt (x * x + y * y) < 0"
    apply (case_tac "x = 0", auto)
    apply (drule_tac y = y in real_sqrt_sum_squares_gt_zero3)
    apply (auto simp add: zero_less_mult_iff divide_inverse power2_eq_square)
    done

    lemma polar_ex1:
         "[| x ≠ 0; 0 < y |] ==> ∃r a. x = r * cos a & y = r * sin a"
    apply (rule_tac x = "sqrt (x² + y²)" in exI)
    apply (rule_tac x = "arcos (x / sqrt (x * x + y * y))" in exI)
    apply auto
    apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
    apply (auto simp add: power2_eq_square)
    apply (simp add: arcos_def)
    apply (cut_tac x1 = x and y1 = y
           in cos_total [OF cos_x_y_ge_minus_one cos_x_y_le_one])
    apply (rule someI2_ex, blast)
    apply (erule_tac V = "EX! xa. 0 ≤ xa & xa ≤ pi & cos xa = x / sqrt (x * x + y * y)" in thin_rl)
    apply (frule sin_x_y_disj, blast)
    apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
    apply (auto simp add: power2_eq_square)
    apply (drule sin_ge_zero, assumption)
    apply (drule_tac x = x in real_sqrt_divide_less_zero, auto)
    done

    lemma real_sum_squares_cancel2a: "x * x = -(y * y) ==> y = (0::real)"
    by (auto intro: real_sum_squares_cancel iff: real_add_eq_0_iff)

    lemma polar_ex2:
         "[| x ≠ 0; y < 0 |] ==> ∃r a. x = r * cos a & y = r * sin a"
    apply (cut_tac x = 0 and y = x in linorder_less_linear, auto)
    apply (rule_tac x = "sqrt (x² + y²)" in exI)
    apply (rule_tac x = "arcsin (y / sqrt (x * x + y * y))" in exI)
    apply (auto dest: real_sum_squares_cancel2a
                simp add: power2_eq_square real_0_le_add_iff real_add_eq_0_iff)
    apply (unfold arcsin_def)
    apply (cut_tac x1 = x and y1 = y
           in sin_total [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2])
    apply (rule someI2_ex, blast)
    apply (erule_tac V = "EX! v. ?P v" in thin_rl)
    apply (cut_tac x=x and y=y in cos_x_y_disj, simp, blast)
    apply (auto simp add: real_0_le_add_iff real_add_eq_0_iff)
    apply (drule cos_ge_zero, force)
    apply (drule_tac x = y in real_sqrt_divide_less_zero)
    apply (auto simp add: add_commute)
    apply (insert polar_ex1 [of x "-y"], simp, clarify)
    apply (rule_tac x = r in exI)
    apply (rule_tac x = "-a" in exI, simp)
    done

    lemma polar_Ex: "∃r a. x = r * cos a & y = r * sin a"
    apply (case_tac "x = 0", auto)
    apply (rule_tac x = y in exI)
    apply (rule_tac x = "pi/2" in exI, auto)
    apply (cut_tac x = 0 and y = y in linorder_less_linear, auto)
    apply (rule_tac [2] x = x in exI)
    apply (rule_tac [2] x = 0 in exI, auto)
    apply (blast intro: polar_ex1 polar_ex2)+
    done

    lemma real_sqrt_ge_abs1 [simp]: "¦x¦ ≤ sqrt (x² + y²)"
    apply (rule_tac n = 1 in realpow_increasing)
    apply (auto simp add: numeral_2_eq_2 [symmetric] power2_abs)
    done

    lemma real_sqrt_ge_abs2 [simp]: "¦y¦ ≤ sqrt (x² + y²)"
    apply (rule real_add_commute [THEN subst])
    apply (rule real_sqrt_ge_abs1)
    done
    declare real_sqrt_ge_abs1 [simp] real_sqrt_ge_abs2 [simp]

    lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
    by (auto intro: real_sqrt_gt_zero)

    lemma real_sqrt_two_ge_zero [simp]: "0 ≤ sqrt 2"
    by (auto intro: real_sqrt_ge_zero)

    lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
    apply (rule order_less_le_trans [of _ "7/5"], simp)
    apply (rule_tac n = 1 in realpow_increasing)
      prefer 3 apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
    apply (simp_all add: numeral_2_eq_2)
    done

    lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
    by (simp add: divide_less_eq mult_compare_simps)

    lemma four_x_squared:
      fixes x::real
      shows "4 * x² = (2 * x)²"
    by (simp add: power2_eq_square)

    text{*Needed for the infinitely close relation over the nonstandard
        complex numbers*}
    lemma lemma_sqrt_hcomplex_capprox:
         "[| 0 < u; x < u/2; y < u/2; 0 ≤ x; 0 ≤ y |] ==> sqrt (x² + y²) < u"
    apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
    apply (erule_tac [2] lemma_real_divide_sqrt_less)
    apply (rule_tac n = 1 in realpow_increasing)
    apply (auto simp add: real_0_le_divide_iff realpow_divide numeral_2_eq_2 [symmetric]
            simp del: realpow_Suc)
    apply (rule_tac t = "u²" in real_sum_of_halves [THEN subst])
    apply (rule add_mono)
    apply (auto simp add: four_x_squared simp del: realpow_Suc intro: power_mono)
    done

    declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp]

    subsection{*A Few Theorems Involving Ln, Derivatives, etc.*}

    lemma lemma_DERIV_ln:
         "DERIV ln z :> l ==> DERIV (%x. exp (ln x)) z :> exp (ln z) * l"
    by (erule DERIV_fun_exp)

    lemma STAR_exp_ln: "0 < z ==> ( *f* (%x. exp (ln x))) z = z"
    apply (cases z)
    apply (auto simp add: starfun star_n_zero_num star_n_less star_n_eq_iff)
    done

    lemma hypreal_add_Infinitesimal_gt_zero:
         "[|e : Infinitesimal; 0 < x |] ==> 0 < hypreal_of_real x + e"
    apply (rule_tac c1 = "-e" in add_less_cancel_right [THEN iffD1])
    apply (auto intro: Infinitesimal_less_SReal)
    done

    lemma NSDERIV_exp_ln_one: "0 < z ==> NSDERIV (%x. exp (ln x)) z :> 1"
    apply (simp add: nsderiv_def NSLIM_def, auto)
    apply (rule ccontr)
    apply (subgoal_tac "0 < hypreal_of_real z + h")
    apply (drule STAR_exp_ln)
    apply (rule_tac [2] hypreal_add_Infinitesimal_gt_zero)
    apply (subgoal_tac "h/h = 1")
    apply (auto simp add: exp_ln_iff [symmetric] simp del: exp_ln_iff)
    done

    lemma DERIV_exp_ln_one: "0 < z ==> DERIV (%x. exp (ln x)) z :> 1"
    by (auto intro: NSDERIV_exp_ln_one simp add: NSDERIV_DERIV_iff [symmetric])

    lemma lemma_DERIV_ln2:
         "[| 0 < z; DERIV ln z :> l |] ==>  exp (ln z) * l = 1"
    apply (rule DERIV_unique)
    apply (rule lemma_DERIV_ln)
    apply (rule_tac [2] DERIV_exp_ln_one, auto)
    done

    lemma lemma_DERIV_ln3:
         "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/(exp (ln z))"
    apply (rule_tac c1 = "exp (ln z)" in real_mult_left_cancel [THEN iffD1])
    apply (auto intro: lemma_DERIV_ln2)
    done

    lemma lemma_DERIV_ln4: "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/z"
    apply (rule_tac t = z in exp_ln_iff [THEN iffD2, THEN subst])
    apply (auto intro: lemma_DERIV_ln3)
    done

    (* need to rename second isCont_inverse *)

    lemma isCont_inv_fun:
         "[| 0 < d; ∀z. ¦z - x¦ ≤ d --> g(f(z)) = z;
             ∀z. ¦z - x¦ ≤ d --> isCont f z |]
          ==> isCont g (f x)"
    apply (simp (no_asm) add: isCont_iff LIM_def)
    apply safe
    apply (drule_tac ?d1.0 = r in real_lbound_gt_zero)
    apply (assumption, safe)
    apply (subgoal_tac "∀z. ¦z - x¦ ≤ e --> (g (f z) = z) ")
    prefer 2 apply force
    apply (subgoal_tac "∀z. ¦z - x¦ ≤ e --> isCont f z")
    prefer 2 apply force
    apply (drule_tac d = e in isCont_inj_range)
    prefer 2 apply (assumption, assumption, safe)
    apply (rule_tac x = ea in exI, auto)
    apply (drule_tac x = "f (x) + xa" and P = "%y. ¦y - f x¦ ≤ ea --> (∃z. ¦z - x¦ ≤ e ∧ f z = y)" in spec)
    apply auto
    apply (drule sym, auto, arith)
    done

    lemma isCont_inv_fun_inv:
         "[| 0 < d;
             ∀z. ¦z - x¦ ≤ d --> g(f(z)) = z;
             ∀z. ¦z - x¦ ≤ d --> isCont f z |]
           ==> ∃e. 0 < e &
                 (∀y. 0 < ¦y - f(x)¦ & ¦y - f(x)¦ < e --> f(g(y)) = y)"
    apply (drule isCont_inj_range)
    prefer 2 apply (assumption, assumption, auto)
    apply (rule_tac x = e in exI, auto)
    apply (rotate_tac 2)
    apply (drule_tac x = y in spec, auto)
    done

    text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
    lemma LIM_fun_gt_zero:
         "[| f -- c --> l; 0 < l |]
             ==> ∃r. 0 < r & (∀x. x ≠ c & ¦c - x¦ < r --> 0 < f x)"
    apply (auto simp add: LIM_def)
    apply (drule_tac x = "l/2" in spec, safe, force)
    apply (rule_tac x = s in exI)
    apply (auto simp only: abs_interval_iff)
    done

    lemma LIM_fun_less_zero:
         "[| f -- c --> l; l < 0 |]
          ==> ∃r. 0 < r & (∀x. x ≠ c & ¦c - x¦ < r --> f x < 0)"
    apply (auto simp add: LIM_def)
    apply (drule_tac x = "-l/2" in spec, safe, force)
    apply (rule_tac x = s in exI)
    apply (auto simp only: abs_interval_iff)
    done

    lemma LIM_fun_not_zero:
         "[| f -- c --> l; l ≠ 0 |]
          ==> ∃r. 0 < r & (∀x. x ≠ c & ¦c - x¦ < r --> f x ≠ 0)"
    apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
    apply (drule LIM_fun_less_zero)
    apply (drule_tac [3] LIM_fun_gt_zero)
    apply force+
    done

    ML
    {*
    val inverse_unique = thm "inverse_unique";
    val real_root_zero = thm "real_root_zero";
    val real_root_pow_pos = thm "real_root_pow_pos";
    val real_root_pow_pos2 = thm "real_root_pow_pos2";
    val real_root_pos = thm "real_root_pos";
    val real_root_pos2 = thm "real_root_pos2";
    val real_root_pos_pos = thm "real_root_pos_pos";
    val real_root_pos_pos_le = thm "real_root_pos_pos_le";
    val real_root_one = thm "real_root_one";
    val root_2_eq = thm "root_2_eq";
    val real_sqrt_zero = thm "real_sqrt_zero";
    val real_sqrt_one = thm "real_sqrt_one";
    val real_sqrt_pow2_iff = thm "real_sqrt_pow2_iff";
    val real_sqrt_pow2 = thm "real_sqrt_pow2";
    val real_sqrt_abs_abs = thm "real_sqrt_abs_abs";
    val real_pow_sqrt_eq_sqrt_pow = thm "real_pow_sqrt_eq_sqrt_pow";
    val real_pow_sqrt_eq_sqrt_abs_pow2 = thm "real_pow_sqrt_eq_sqrt_abs_pow2";
    val real_sqrt_pow_abs = thm "real_sqrt_pow_abs";
    val not_real_square_gt_zero = thm "not_real_square_gt_zero";
    val real_sqrt_gt_zero = thm "real_sqrt_gt_zero";
    val real_sqrt_ge_zero = thm "real_sqrt_ge_zero";
    val sqrt_eqI = thm "sqrt_eqI";
    val real_sqrt_mult_distrib = thm "real_sqrt_mult_distrib";
    val real_sqrt_mult_distrib2 = thm "real_sqrt_mult_distrib2";
    val real_sqrt_sum_squares_ge_zero = thm "real_sqrt_sum_squares_ge_zero";
    val real_sqrt_sum_squares_mult_ge_zero = thm "real_sqrt_sum_squares_mult_ge_zero";
    val real_sqrt_sum_squares_mult_squared_eq = thm "real_sqrt_sum_squares_mult_squared_eq";
    val real_sqrt_abs = thm "real_sqrt_abs";
    val real_sqrt_abs2 = thm "real_sqrt_abs2";
    val real_sqrt_pow2_gt_zero = thm "real_sqrt_pow2_gt_zero";
    val real_sqrt_not_eq_zero = thm "real_sqrt_not_eq_zero";
    val real_inv_sqrt_pow2 = thm "real_inv_sqrt_pow2";
    val real_sqrt_eq_zero_cancel = thm "real_sqrt_eq_zero_cancel";
    val real_sqrt_eq_zero_cancel_iff = thm "real_sqrt_eq_zero_cancel_iff";
    val real_sqrt_sum_squares_ge1 = thm "real_sqrt_sum_squares_ge1";
    val real_sqrt_sum_squares_ge2 = thm "real_sqrt_sum_squares_ge2";
    val real_sqrt_ge_one = thm "real_sqrt_ge_one";
    val summable_exp = thm "summable_exp";
    val summable_sin = thm "summable_sin";
    val summable_cos = thm "summable_cos";
    val exp_converges = thm "exp_converges";
    val sin_converges = thm "sin_converges";
    val cos_converges = thm "cos_converges";
    val powser_insidea = thm "powser_insidea";
    val powser_inside = thm "powser_inside";
    val diffs_minus = thm "diffs_minus";
    val diffs_equiv = thm "diffs_equiv";
    val less_add_one = thm "less_add_one";
    val termdiffs_aux = thm "termdiffs_aux";
    val termdiffs = thm "termdiffs";
    val exp_fdiffs = thm "exp_fdiffs";
    val sin_fdiffs = thm "sin_fdiffs";
    val sin_fdiffs2 = thm "sin_fdiffs2";
    val cos_fdiffs = thm "cos_fdiffs";
    val cos_fdiffs2 = thm "cos_fdiffs2";
    val DERIV_exp = thm "DERIV_exp";
    val DERIV_sin = thm "DERIV_sin";
    val DERIV_cos = thm "DERIV_cos";
    val exp_zero = thm "exp_zero";
    (* val exp_ge_add_one_self = thm "exp_ge_add_one_self"; *)
    val exp_gt_one = thm "exp_gt_one";
    val DERIV_exp_add_const = thm "DERIV_exp_add_const";
    val DERIV_exp_minus = thm "DERIV_exp_minus";
    val DERIV_exp_exp_zero = thm "DERIV_exp_exp_zero";
    val exp_add_mult_minus = thm "exp_add_mult_minus";
    val exp_mult_minus = thm "exp_mult_minus";
    val exp_mult_minus2 = thm "exp_mult_minus2";
    val exp_minus = thm "exp_minus";
    val exp_add = thm "exp_add";
    val exp_ge_zero = thm "exp_ge_zero";
    val exp_not_eq_zero = thm "exp_not_eq_zero";
    val exp_gt_zero = thm "exp_gt_zero";
    val inv_exp_gt_zero = thm "inv_exp_gt_zero";
    val abs_exp_cancel = thm "abs_exp_cancel";
    val exp_real_of_nat_mult = thm "exp_real_of_nat_mult";
    val exp_diff = thm "exp_diff";
    val exp_less_mono = thm "exp_less_mono";
    val exp_less_cancel = thm "exp_less_cancel";
    val exp_less_cancel_iff = thm "exp_less_cancel_iff";
    val exp_le_cancel_iff = thm "exp_le_cancel_iff";
    val exp_inj_iff = thm "exp_inj_iff";
    val exp_total = thm "exp_total";
    val ln_exp = thm "ln_exp";
    val exp_ln_iff = thm "exp_ln_iff";
    val ln_mult = thm "ln_mult";
    val ln_inj_iff = thm "ln_inj_iff";
    val ln_one = thm "ln_one";
    val ln_inverse = thm "ln_inverse";
    val ln_div = thm "ln_div";
    val ln_less_cancel_iff = thm "ln_less_cancel_iff";
    val ln_le_cancel_iff = thm "ln_le_cancel_iff";
    val ln_realpow = thm "ln_realpow";
    val ln_add_one_self_le_self = thm "ln_add_one_self_le_self";
    val ln_less_self = thm "ln_less_self";
    val ln_ge_zero = thm "ln_ge_zero";
    val ln_gt_zero = thm "ln_gt_zero";
    val ln_less_zero = thm "ln_less_zero";
    val exp_ln_eq = thm "exp_ln_eq";
    val sin_zero = thm "sin_zero";
    val cos_zero = thm "cos_zero";
    val DERIV_sin_sin_mult = thm "DERIV_sin_sin_mult";
    val DERIV_sin_sin_mult2 = thm "DERIV_sin_sin_mult2";
    val DERIV_sin_realpow2 = thm "DERIV_sin_realpow2";
    val DERIV_sin_realpow2a = thm "DERIV_sin_realpow2a";
    val DERIV_cos_cos_mult = thm "DERIV_cos_cos_mult";
    val DERIV_cos_cos_mult2 = thm "DERIV_cos_cos_mult2";
    val DERIV_cos_realpow2 = thm "DERIV_cos_realpow2";
    val DERIV_cos_realpow2a = thm "DERIV_cos_realpow2a";
    val DERIV_cos_realpow2b = thm "DERIV_cos_realpow2b";
    val DERIV_cos_cos_mult3 = thm "DERIV_cos_cos_mult3";
    val DERIV_sin_circle_all = thm "DERIV_sin_circle_all";
    val DERIV_sin_circle_all_zero = thm "DERIV_sin_circle_all_zero";
    val sin_cos_squared_add = thm "sin_cos_squared_add";
    val sin_cos_squared_add2 = thm "sin_cos_squared_add2";
    val sin_cos_squared_add3 = thm "sin_cos_squared_add3";
    val sin_squared_eq = thm "sin_squared_eq";
    val cos_squared_eq = thm "cos_squared_eq";
    val real_gt_one_ge_zero_add_less = thm "real_gt_one_ge_zero_add_less";
    val abs_sin_le_one = thm "abs_sin_le_one";
    val sin_ge_minus_one = thm "sin_ge_minus_one";
    val sin_le_one = thm "sin_le_one";
    val abs_cos_le_one = thm "abs_cos_le_one";
    val cos_ge_minus_one = thm "cos_ge_minus_one";
    val cos_le_one = thm "cos_le_one";
    val DERIV_fun_pow = thm "DERIV_fun_pow";
    val DERIV_fun_exp = thm "DERIV_fun_exp";
    val DERIV_fun_sin = thm "DERIV_fun_sin";
    val DERIV_fun_cos = thm "DERIV_fun_cos";
    val DERIV_intros = thms "DERIV_intros";
    val sin_cos_add = thm "sin_cos_add";
    val sin_add = thm "sin_add";
    val cos_add = thm "cos_add";
    val sin_cos_minus = thm "sin_cos_minus";
    val sin_minus = thm "sin_minus";
    val cos_minus = thm "cos_minus";
    val sin_diff = thm "sin_diff";
    val sin_diff2 = thm "sin_diff2";
    val cos_diff = thm "cos_diff";
    val cos_diff2 = thm "cos_diff2";
    val sin_double = thm "sin_double";
    val cos_double = thm "cos_double";
    val sin_paired = thm "sin_paired";
    val sin_gt_zero = thm "sin_gt_zero";
    val sin_gt_zero1 = thm "sin_gt_zero1";
    val cos_double_less_one = thm "cos_double_less_one";
    val cos_paired = thm "cos_paired";
    val cos_two_less_zero = thm "cos_two_less_zero";
    val cos_is_zero = thm "cos_is_zero";
    val pi_half = thm "pi_half";
    val cos_pi_half = thm "cos_pi_half";
    val pi_half_gt_zero = thm "pi_half_gt_zero";
    val pi_half_less_two = thm "pi_half_less_two";
    val pi_gt_zero = thm "pi_gt_zero";
    val pi_neq_zero = thm "pi_neq_zero";
    val pi_not_less_zero = thm "pi_not_less_zero";
    val pi_ge_zero = thm "pi_ge_zero";
    val minus_pi_half_less_zero = thm "minus_pi_half_less_zero";
    val sin_pi_half = thm "sin_pi_half";
    val cos_pi = thm "cos_pi";
    val sin_pi = thm "sin_pi";
    val sin_cos_eq = thm "sin_cos_eq";
    val minus_sin_cos_eq = thm "minus_sin_cos_eq";
    val cos_sin_eq = thm "cos_sin_eq";
    val sin_periodic_pi = thm "sin_periodic_pi";
    val sin_periodic_pi2 = thm "sin_periodic_pi2";
    val cos_periodic_pi = thm "cos_periodic_pi";
    val sin_periodic = thm "sin_periodic";
    val cos_periodic = thm "cos_periodic";
    val cos_npi = thm "cos_npi";
    val sin_npi = thm "sin_npi";
    val sin_npi2 = thm "sin_npi2";
    val cos_two_pi = thm "cos_two_pi";
    val sin_two_pi = thm "sin_two_pi";
    val sin_gt_zero2 = thm "sin_gt_zero2";
    val sin_less_zero = thm "sin_less_zero";
    val pi_less_4 = thm "pi_less_4";
    val cos_gt_zero = thm "cos_gt_zero";
    val cos_gt_zero_pi = thm "cos_gt_zero_pi";
    val cos_ge_zero = thm "cos_ge_zero";
    val sin_gt_zero_pi = thm "sin_gt_zero_pi";
    val sin_ge_zero = thm "sin_ge_zero";
    val cos_total = thm "cos_total";
    val sin_total = thm "sin_total";
    val reals_Archimedean4 = thm "reals_Archimedean4";
    val cos_zero_lemma = thm "cos_zero_lemma";
    val sin_zero_lemma = thm "sin_zero_lemma";
    val cos_zero_iff = thm "cos_zero_iff";
    val sin_zero_iff = thm "sin_zero_iff";
    val tan_zero = thm "tan_zero";
    val tan_pi = thm "tan_pi";
    val tan_npi = thm "tan_npi";
    val tan_minus = thm "tan_minus";
    val tan_periodic = thm "tan_periodic";
    val add_tan_eq = thm "add_tan_eq";
    val tan_add = thm "tan_add";
    val tan_double = thm "tan_double";
    val tan_gt_zero = thm "tan_gt_zero";
    val tan_less_zero = thm "tan_less_zero";
    val DERIV_tan = thm "DERIV_tan";
    val LIM_cos_div_sin = thm "LIM_cos_div_sin";
    val tan_total_pos = thm "tan_total_pos";
    val tan_total = thm "tan_total";
    val arcsin_pi = thm "arcsin_pi";
    val arcsin = thm "arcsin";
    val sin_arcsin = thm "sin_arcsin";
    val arcsin_bounded = thm "arcsin_bounded";
    val arcsin_lbound = thm "arcsin_lbound";
    val arcsin_ubound = thm "arcsin_ubound";
    val arcsin_lt_bounded = thm "arcsin_lt_bounded";
    val arcsin_sin = thm "arcsin_sin";
    val arcos = thm "arcos";
    val cos_arcos = thm "cos_arcos";
    val arcos_bounded = thm "arcos_bounded";
    val arcos_lbound = thm "arcos_lbound";
    val arcos_ubound = thm "arcos_ubound";
    val arcos_lt_bounded = thm "arcos_lt_bounded";
    val arcos_cos = thm "arcos_cos";
    val arcos_cos2 = thm "arcos_cos2";
    val arctan = thm "arctan";
    val tan_arctan = thm "tan_arctan";
    val arctan_bounded = thm "arctan_bounded";
    val arctan_lbound = thm "arctan_lbound";
    val arctan_ubound = thm "arctan_ubound";
    val arctan_tan = thm "arctan_tan";
    val arctan_zero_zero = thm "arctan_zero_zero";
    val cos_arctan_not_zero = thm "cos_arctan_not_zero";
    val tan_sec = thm "tan_sec";
    val DERIV_sin_add = thm "DERIV_sin_add";
    val cos_2npi = thm "cos_2npi";
    val cos_3over2_pi = thm "cos_3over2_pi";
    val sin_2npi = thm "sin_2npi";
    val sin_3over2_pi = thm "sin_3over2_pi";
    val cos_pi_eq_zero = thm "cos_pi_eq_zero";
    val DERIV_cos_add = thm "DERIV_cos_add";
    val isCont_cos = thm "isCont_cos";
    val isCont_sin = thm "isCont_sin";
    val isCont_exp = thm "isCont_exp";
    val sin_zero_abs_cos_one = thm "sin_zero_abs_cos_one";
    val exp_eq_one_iff = thm "exp_eq_one_iff";
    val cos_one_sin_zero = thm "cos_one_sin_zero";
    val real_root_less_mono = thm "real_root_less_mono";
    val real_root_le_mono = thm "real_root_le_mono";
    val real_root_less_iff = thm "real_root_less_iff";
    val real_root_le_iff = thm "real_root_le_iff";
    val real_root_eq_iff = thm "real_root_eq_iff";
    val real_root_pos_unique = thm "real_root_pos_unique";
    val real_root_mult = thm "real_root_mult";
    val real_root_inverse = thm "real_root_inverse";
    val real_root_divide = thm "real_root_divide";
    val real_sqrt_less_mono = thm "real_sqrt_less_mono";
    val real_sqrt_le_mono = thm "real_sqrt_le_mono";
    val real_sqrt_less_iff = thm "real_sqrt_less_iff";
    val real_sqrt_le_iff = thm "real_sqrt_le_iff";
    val real_sqrt_eq_iff = thm "real_sqrt_eq_iff";
    val real_sqrt_sos_less_one_iff = thm "real_sqrt_sos_less_one_iff";
    val real_sqrt_sos_eq_one_iff = thm "real_sqrt_sos_eq_one_iff";
    val real_divide_square_eq = thm "real_divide_square_eq";
    val real_sqrt_sum_squares_gt_zero1 = thm "real_sqrt_sum_squares_gt_zero1";
    val real_sqrt_sum_squares_gt_zero2 = thm "real_sqrt_sum_squares_gt_zero2";
    val real_sqrt_sum_squares_gt_zero3 = thm "real_sqrt_sum_squares_gt_zero3";
    val real_sqrt_sum_squares_gt_zero3a = thm "real_sqrt_sum_squares_gt_zero3a";
    val cos_x_y_ge_minus_one = thm "cos_x_y_ge_minus_one";
    val cos_x_y_ge_minus_one1a = thm "cos_x_y_ge_minus_one1a";
    val cos_x_y_le_one = thm "cos_x_y_le_one";
    val cos_x_y_le_one2 = thm "cos_x_y_le_one2";
    val cos_abs_x_y_ge_minus_one = thm "cos_abs_x_y_ge_minus_one";
    val cos_abs_x_y_le_one = thm "cos_abs_x_y_le_one";
    val minus_pi_less_zero = thm "minus_pi_less_zero";
    val arcos_ge_minus_pi = thm "arcos_ge_minus_pi";
    val sin_x_y_disj = thm "sin_x_y_disj";
    val cos_x_y_disj = thm "cos_x_y_disj";
    val real_sqrt_divide_less_zero = thm "real_sqrt_divide_less_zero";
    val polar_ex1 = thm "polar_ex1";
    val polar_ex2 = thm "polar_ex2";
    val polar_Ex = thm "polar_Ex";
    val real_sqrt_ge_abs1 = thm "real_sqrt_ge_abs1";
    val real_sqrt_ge_abs2 = thm "real_sqrt_ge_abs2";
    val real_sqrt_two_gt_zero = thm "real_sqrt_two_gt_zero";
    val real_sqrt_two_ge_zero = thm "real_sqrt_two_ge_zero";
    val real_sqrt_two_gt_one = thm "real_sqrt_two_gt_one";
    val STAR_exp_ln = thm "STAR_exp_ln";
    val hypreal_add_Infinitesimal_gt_zero = thm "hypreal_add_Infinitesimal_gt_zero";
    val NSDERIV_exp_ln_one = thm "NSDERIV_exp_ln_one";
    val DERIV_exp_ln_one = thm "DERIV_exp_ln_one";
    val isCont_inv_fun = thm "isCont_inv_fun";
    val isCont_inv_fun_inv = thm "isCont_inv_fun_inv";
    val LIM_fun_gt_zero = thm "LIM_fun_gt_zero";
    val LIM_fun_less_zero = thm "LIM_fun_less_zero";
    val LIM_fun_not_zero = thm "LIM_fun_not_zero";
    *}

    end

lemma real_root_zero:

      root (Suc n) 0 = 0

lemma real_root_pow_pos:

      0 < x ==> root (Suc n) x ^ Suc n = x

lemma real_root_pow_pos2:

      0 ≤ x ==> root (Suc n) x ^ Suc n = x

lemma real_root_pos:

      0 < x ==> root (Suc n) (x ^ Suc n) = x

lemma real_root_pos2:

      0 ≤ x ==> root (Suc n) (x ^ Suc n) = x

lemma real_root_pos_pos:

      0 < x ==> 0 ≤ root (Suc n) x

lemma real_root_pos_pos_le:

      0 ≤ x ==> 0 ≤ root (Suc n) x

lemma real_root_one:

      root (Suc n) 1 = 1

### Square Root

lemma root_2_eq:

      root 2 = root (Suc (Suc 0))

lemma real_sqrt_zero:

      sqrt 0 = 0

lemma real_sqrt_one:

      sqrt 1 = 1

lemma real_sqrt_pow2_iff:

      ((sqrt x)² = x) = (0 ≤ x)

lemma

      (sqrt (u2.0² + v2.0²))² = u2.0² + v2.0²

lemma real_sqrt_pow2:

      0 ≤ x ==> (sqrt x)² = x

lemma real_sqrt_abs_abs:

      (sqrt ¦x¦)² = ¦x¦

lemma real_pow_sqrt_eq_sqrt_pow:

      0 ≤ x ==> (sqrt x)² = sqrt (x²)

lemma real_pow_sqrt_eq_sqrt_abs_pow2:

      0 ≤ x ==> (sqrt x)² = sqrt (¦x¦²)

lemma real_sqrt_pow_abs:

      0 ≤ x ==> (sqrt x)² = ¦x¦

lemma not_real_square_gt_zero:

      (¬ 0 < x * x) = (x = 0)

lemma real_sqrt_gt_zero:

      0 < x ==> 0 < sqrt x

lemma real_sqrt_ge_zero:

      0 ≤ x ==> 0 ≤ sqrt x

lemma real_sqrt_mult_self_sum_ge_zero:

      0 ≤ sqrt (x * x + y * y)

lemma sqrt_eqI:

      [| r² = a; 0 ≤ r |] ==> sqrt a = r

lemma real_sqrt_mult_distrib:

      [| 0 ≤ x; 0 ≤ y |] ==> sqrt (x * y) = sqrt x * sqrt y

lemma real_sqrt_mult_distrib2:

      [| 0 ≤ x; 0 ≤ y |] ==> sqrt (x * y) = sqrt x * sqrt y

lemma real_sqrt_sum_squares_ge_zero:

      0 ≤ sqrt (x² + y²)

lemma real_sqrt_sum_squares_mult_ge_zero:

      0 ≤ sqrt ((x² + y²) * (xa² + ya²))

lemma real_sqrt_sum_squares_mult_squared_eq:

      (sqrt ((x² + y²) * (xa² + ya²)))² = (x² + y²) * (xa² + ya²)

lemma real_sqrt_abs:

      sqrt (x²) = ¦x¦

lemma real_sqrt_abs2:

      sqrt (x * x) = ¦x¦

lemma real_sqrt_pow2_gt_zero:

      0 < x ==> 0 < (sqrt x)²

lemma real_sqrt_not_eq_zero:

      0 < x ==> sqrt x ≠ 0

lemma real_inv_sqrt_pow2:

      0 < x ==> (inverse (sqrt x))² = inverse x

lemma real_sqrt_eq_zero_cancel:

      [| 0 ≤ x; sqrt x = 0 |] ==> x = 0

lemma real_sqrt_eq_zero_cancel_iff:

      0 ≤ x ==> (sqrt x = 0) = (x = 0)

lemma real_sqrt_sum_squares_ge1:

      x ≤ sqrt (x² + y²)

lemma real_sqrt_sum_squares_ge2:

      y ≤ sqrt (z² + y²)

lemma real_sqrt_ge_one:

      1 ≤ x ==> 1 ≤ sqrt x

### Exponential Function

lemma summable_exp:

      summable (%n. inverse (real (fact n)) * x ^ n)

lemma summable_sin:

      summable
       (%n. (if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
            x ^ n)

lemma summable_cos:

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

lemma lemma_STAR_sin:

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

lemma lemma_STAR_cos:

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

lemma lemma_STAR_cos1:

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

lemma lemma_STAR_cos2:

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

lemma exp_converges:

      (%n. inverse (real (fact n)) * x ^ n) sums exp x

lemma sin_converges:

      (%n. (if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
           x ^ n) sums
      sin x

lemma cos_converges:

      (%n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) * x ^ n) sums
      cos x

lemma lemma_realpow_diff:

      p ≤ n ==> y ^ (Suc n - p) = y ^ (n - p) * y

### Properties of Power Series

lemma lemma_realpow_diff_sumr:

      (∑p = 0..<Suc n. x ^ p * y ^ (Suc n - p)) =
      y * (∑p = 0..<Suc n. x ^ p * y ^ (n - p))

lemma lemma_realpow_diff_sumr2:

      x ^ Suc n - y ^ Suc n = (x - y) * (∑p = 0..<Suc n. x ^ p * y ^ (n - p))

lemma lemma_realpow_rev_sumr:

      (∑p = 0..<Suc n. x ^ p * y ^ (n - p)) = (∑p = 0..<Suc n. x ^ (n - p) * y ^ p)

lemma powser_insidea:

      [| summable (%n. f n * x ^ n); ¦z¦ < ¦x¦ |] ==> summable (%n. ¦f n¦ * z ^ n)

lemma powser_inside:

      [| summable (%n. f n * x ^ n); ¦z¦ < ¦x¦ |] ==> summable (%n. f n * z ^ n)

### Differentiation of Power Series

lemma diffs_minus:

      diffs (%n. - c n) = (%n. - diffs c n)

lemma lemma_diffs:

      (∑n = 0..<n. diffs c n * x ^ n) =
      (∑n = 0..<n. real n * c n * x ^ (n - Suc 0)) + real n * c n * x ^ (n - Suc 0)

lemma lemma_diffs2:

      (∑n = 0..<n. real n * c n * x ^ (n - Suc 0)) =
      (∑n = 0..<n. diffs c n * x ^ n) - real n * c n * x ^ (n - Suc 0)

lemma diffs_equiv:

      summable (%n. diffs c n * x ^ n)
      ==> (%n. real n * c n * x ^ (n - Suc 0)) sums (∑n. diffs c n * x ^ n)

### Term-by-Term Differentiability of Power Series

lemma lemma_termdiff1:

      (∑p = 0..<m. (z + h) ^ (m - p) * z ^ p - z ^ m) =
      (∑p = 0..<m. z ^ p * ((z + h) ^ (m - p) - z ^ (m - p)))

lemma less_add_one:

      m < n ==> ∃d. n = m + d + Suc 0

lemma sumdiff:

      a + b - (c + d) = a - c + b - d

lemma lemma_termdiff2:

      h ≠ 0
      ==> ((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0) =
          h * (∑p = 0..<n - Suc 0.
                  z ^ p * (∑q = 0..<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - p - q)))

lemma lemma_termdiff3:

      [| h ≠ 0; ¦z¦ ≤ K; ¦z + h¦ ≤ K |]
      ==> ¦((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0)¦
          ≤ real n * real (n - Suc 0) * K ^ (n - 2) * ¦h¦

lemma lemma_termdiff4:

      [| 0 < k; ∀h. 0 < ¦h¦ ∧ ¦h¦ < k --> ¦f h¦ ≤ K * ¦h¦ |] ==> f -- 0 --> 0

lemma lemma_termdiff5:

      [| 0 < k; summable f; ∀h. 0 < ¦h¦ ∧ ¦h¦ < k --> (∀n. ¦g h n¦ ≤ f n * ¦h¦) |]
      ==> (%h. suminf (g h)) -- 0 --> 0

lemma termdiffs_aux:

      [| summable (%n. diffs (diffs c) n * K ^ n); ¦x¦ < ¦K¦ |]
      ==> (%h. ∑n. c n *
                   (((x + h) ^ n - x ^ n) * inverse h - real n * x ^ (n - Suc 0)))
          -- 0 --> 0

lemma termdiffs:

      [| summable (%n. c n * K ^ n); summable (%n. diffs c n * K ^ n);
         summable (%n. diffs (diffs c) n * K ^ n); ¦x¦ < ¦K¦ |]
      ==> DERIV (%x. ∑n. c n * x ^ n) x :> (∑n. diffs c n * x ^ n)

### Formal Derivatives of Exp, Sin, and Cos Series

lemma exp_fdiffs:

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

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)

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)

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)))

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))

lemma lemma_sin_minus:

      - sin x =
      (∑n. - ((if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
              x ^ n))

lemma lemma_exp_ext:

      exp = (%x. ∑n. inverse (real (fact n)) * x ^ n)

lemma DERIV_exp:

      DERIV exp x :> exp x

lemma lemma_sin_ext:

      sin =
      (%x. ∑n. (if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
               x ^ n)

lemma lemma_cos_ext:

      cos =
      (%x. ∑n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) * x ^ n)

lemma DERIV_sin:

      DERIV sin x :> cos x

lemma DERIV_cos:

      DERIV cos x :> - sin x

### Properties of the Exponential Function

lemma exp_zero:

      exp 0 = 1

lemma exp_ge_add_one_self_aux:

      0 ≤ x ==> 1 + x ≤ exp x

lemma exp_gt_one:

      0 < x ==> 1 < exp x

lemma DERIV_exp_add_const:

      DERIV (%x. exp (x + y)) x :> exp (x + y)

lemma DERIV_exp_minus:

      DERIV (%x. exp (- x)) x :> - exp (- x)

lemma DERIV_exp_exp_zero:

      DERIV (%x. exp (x + y) * exp (- x)) x :> 0

lemma exp_add_mult_minus:

      exp (x + y) * exp (- x) = exp y

lemma exp_mult_minus:

      exp x * exp (- x) = 1

lemma exp_mult_minus2:

      exp (- x) * exp x = 1

lemma exp_minus:

      exp (- x) = inverse (exp x)

lemma exp_add:

      exp (x + y) = exp x * exp y

lemma exp_ge_zero:

      0 ≤ exp x

lemma exp_not_eq_zero:

      exp x ≠ 0

lemma exp_gt_zero:

      0 < exp x

lemma inv_exp_gt_zero:

      0 < inverse (exp x)

lemma abs_exp_cancel:

      ¦exp x¦ = exp x

lemma exp_real_of_nat_mult:

      exp (real n * x) = exp x ^ n

lemma exp_diff:

      exp (x - y) = exp x / exp y

lemma exp_less_mono:

      x < y ==> exp x < exp y

lemma exp_less_cancel:

      exp x < exp y ==> x < y

lemma exp_less_cancel_iff:

      (exp x < exp y) = (x < y)

lemma exp_le_cancel_iff:

      (exp x ≤ exp y) = (x ≤ y)

lemma exp_inj_iff:

      (exp x = exp y) = (x = y)

lemma lemma_exp_total:

      1 ≤ y ==> ∃x≥0. x ≤ y - 1 ∧ exp x = y

lemma exp_total:

      0 < y ==> ∃x. exp x = y

### Properties of the Logarithmic Function

lemma ln_exp:

      ln (exp x) = x

lemma exp_ln_iff:

      (exp (ln x) = x) = (0 < x)

lemma ln_mult:

      [| 0 < x; 0 < y |] ==> ln (x * y) = ln x + ln y

lemma ln_inj_iff:

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

lemma ln_one:

      ln 1 = 0

lemma ln_inverse:

      0 < x ==> ln (inverse x) = - ln x

lemma ln_div:

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

lemma ln_less_cancel_iff:

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

lemma ln_le_cancel_iff:

      [| 0 < x; 0 < y |] ==> (ln x ≤ ln y) = (x ≤ y)

lemma ln_realpow:

      0 < x ==> ln (x ^ n) = real n * ln x

lemma ln_add_one_self_le_self:

      0 ≤ x ==> ln (1 + x) ≤ x

lemma ln_less_self:

      0 < x ==> ln x < x

lemma ln_ge_zero:

      1 ≤ x ==> 0 ≤ ln x

lemma ln_ge_zero_imp_ge_one:

      [| 0 ≤ ln x; 0 < x |] ==> 1 ≤ x

lemma ln_ge_zero_iff:

      0 < x ==> (0 ≤ ln x) = (1 ≤ x)

lemma ln_less_zero_iff:

      0 < x ==> (ln x < 0) = (x < 1)

lemma ln_gt_zero:

      1 < x ==> 0 < ln x

lemma ln_gt_zero_imp_gt_one:

      [| 0 < ln x; 0 < x |] ==> 1 < x

lemma ln_gt_zero_iff:

      0 < x ==> (0 < ln x) = (1 < x)

lemma ln_eq_zero_iff:

      0 < x ==> (ln x = 0) = (x = 1)

lemma ln_less_zero:

      [| 0 < x; x < 1 |] ==> ln x < 0

lemma exp_ln_eq:

      exp u = x ==> ln x = u

### Basic Properties of the Trigonometric Functions

lemma sin_zero:

      sin 0 = 0

lemma lemma_series_zero2:

      (∀m≥n. f m = 0) --> f sums setsum f {0..<n}

lemma cos_zero:

      cos 0 = 1

lemma DERIV_sin_sin_mult:

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

lemma DERIV_sin_sin_mult2:

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

lemma DERIV_sin_realpow2:

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

lemma DERIV_sin_realpow2a:

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

lemma DERIV_cos_cos_mult:

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

lemma DERIV_cos_cos_mult2:

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

lemma DERIV_cos_realpow2:

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

lemma DERIV_cos_realpow2a:

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

lemma lemma_DERIV_subst:

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

lemma DERIV_cos_realpow2b:

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

lemma DERIV_cos_cos_mult3:

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

lemma DERIV_sin_circle_all:

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

lemma DERIV_sin_circle_all_zero:

      ∀x. DERIV (%x. (sin x)² + (cos x)²) x :> 0

lemma sin_cos_squared_add:

      (sin x)² + (cos x)² = 1

lemma sin_cos_squared_add2:

      (cos x)² + (sin x)² = 1

lemma sin_cos_squared_add3:

      cos x * cos x + sin x * sin x = 1

lemma sin_squared_eq:

      (sin x)² = 1 - (cos x)²

lemma cos_squared_eq:

      (cos x)² = 1 - (sin x)²

lemma real_gt_one_ge_zero_add_less:

      [| 1 < x; 0 ≤ y |] ==> 1 < x + y

lemma abs_sin_le_one:

      ¦sin x¦ ≤ 1

lemma sin_ge_minus_one:

      -1 ≤ sin x

lemma sin_le_one:

      sin x ≤ 1

lemma abs_cos_le_one:

      ¦cos x¦ ≤ 1

lemma cos_ge_minus_one:

      -1 ≤ cos x

lemma cos_le_one:

      cos x ≤ 1

lemma DERIV_fun_pow:

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

lemma DERIV_fun_exp:

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

lemma DERIV_fun_sin:

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

lemma DERIV_fun_cos:

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

lemmas DERIV_intros:

      DERIV (%x. x) x :> 1

      DERIV (%x. k) x :> 0

      DERIV cos x :> - sin x

      DERIV f x :> D ==> DERIV (%x. c * f x) x :> c * D

      DERIV sin x :> cos x

      DERIV exp x :> exp x

      x ≠ 0 ==> DERIV inverse x :> - (inverse x ^ Suc (Suc 0))

      DERIV (%x. x ^ n) x :> real n * x ^ (n - Suc 0)

      [| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x + g x) x :> Da + Db

      [| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x - g x) x :> Da - Db

      [| DERIV f x :> Da; DERIV g x :> Db |]
      ==> DERIV (%x. f x * g x) x :> Da * g x + Db * f x

      DERIV f x :> D ==> DERIV (%x. - f x) x :> - D

      [| DERIV f x :> d; f x ≠ 0 |]
      ==> DERIV (%x. inverse (f x)) x :> - (d * inverse (f x ^ Suc (Suc 0)))

      [| DERIV f x :> d; DERIV g x :> e; g x ≠ 0 |]
      ==> DERIV (%y. f y / g y) x :> (d * g x + - (e * f x)) / g x ^ Suc (Suc 0)

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

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

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

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

lemmas DERIV_intros:

      DERIV (%x. x) x :> 1

      DERIV (%x. k) x :> 0

      DERIV cos x :> - sin x

      DERIV f x :> D ==> DERIV (%x. c * f x) x :> c * D

      DERIV sin x :> cos x

      DERIV exp x :> exp x

      x ≠ 0 ==> DERIV inverse x :> - (inverse x ^ Suc (Suc 0))

      DERIV (%x. x ^ n) x :> real n * x ^ (n - Suc 0)

      [| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x + g x) x :> Da + Db

      [| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x - g x) x :> Da - Db

      [| DERIV f x :> Da; DERIV g x :> Db |]
      ==> DERIV (%x. f x * g x) x :> Da * g x + Db * f x

      DERIV f x :> D ==> DERIV (%x. - f x) x :> - D

      [| DERIV f x :> d; f x ≠ 0 |]
      ==> DERIV (%x. inverse (f x)) x :> - (d * inverse (f x ^ Suc (Suc 0)))

      [| DERIV f x :> d; DERIV g x :> e; g x ≠ 0 |]
      ==> DERIV (%y. f y / g y) x :> (d * g x + - (e * f x)) / g x ^ Suc (Suc 0)

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

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

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

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

lemma lemma_DERIV_sin_cos_add:

      ∀x. DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y))² +
                     (cos (x + y) - (cos x * cos y - sin x * sin y))²)
          x :> 0

lemma sin_cos_add:

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

lemma sin_add:

      sin (x + y) = sin x * cos y + cos x * sin y

lemma cos_add:

      cos (x + y) = cos x * cos y - sin x * sin y

lemma lemma_DERIV_sin_cos_minus:

      ∀x. DERIV (%x. (sin (- x) + sin x)² + (cos (- x) - cos x)²) x :> 0

lemma sin_cos_minus:

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

lemma sin_minus:

      sin (- x) = - sin x

lemma cos_minus:

      cos (- x) = cos x

lemma sin_diff:

      sin (x - y) = sin x * cos y - cos x * sin y

lemma sin_diff2:

      sin (x - y) = cos y * sin x - sin y * cos x

lemma cos_diff:

      cos (x - y) = cos x * cos y + sin x * sin y

lemma cos_diff2:

      cos (x - y) = cos y * cos x + sin y * sin x

lemma sin_double:

      sin (2 * x) = 2 * sin x * cos x

lemma cos_double:

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

### The Constant Pi

lemma sin_paired:

      (%n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums sin x

lemma sin_gt_zero:

      [| 0 < x; x < 2 |] ==> 0 < sin x

lemma sin_gt_zero1:

      [| 0 < x; x < 2 |] ==> 0 < sin x

lemma cos_double_less_one:

      [| 0 < x; x < 2 |] ==> cos (2 * x) < 1

lemma cos_paired:

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

lemma fact_lemma:

      real n * 4 = real (4 * n)

lemma cos_two_less_zero:

      cos 2 < 0

lemma cos_is_zero:

      ∃!x. 0 ≤ x ∧ x ≤ 2 ∧ cos x = 0

lemma pi_half:

      pi / 2 = (SOME x. 0 ≤ x ∧ x ≤ 2 ∧ cos x = 0)

lemma cos_pi_half:

      cos (pi / 2) = 0

lemma pi_half_gt_zero:

      0 < pi / 2

lemma pi_half_less_two:

      pi / 2 < 2

lemma pi_gt_zero:

      0 < pi

lemma pi_neq_zero:

      pi ≠ 0

lemma pi_not_less_zero:

      ¬ pi < 0

lemma pi_ge_zero:

      0 ≤ pi

lemma minus_pi_half_less_zero:

      - (pi / 2) < 0

lemma sin_pi_half:

      sin (pi / 2) = 1

lemma cos_pi:

      cos pi = -1

lemma sin_pi:

      sin pi = 0

lemma sin_cos_eq:

      sin x = cos (pi / 2 - x)

lemma minus_sin_cos_eq:

      - sin x = cos (x + pi / 2)

lemma cos_sin_eq:

      cos x = sin (pi / 2 - x)

lemma sin_periodic_pi:

      sin (x + pi) = - sin x

lemma sin_periodic_pi2:

      sin (pi + x) = - sin x

lemma cos_periodic_pi:

      cos (x + pi) = - cos x

lemma sin_periodic:

      sin (x + 2 * pi) = sin x

lemma cos_periodic:

      cos (x + 2 * pi) = cos x

lemma cos_npi:

      cos (real n * pi) = -1 ^ n

lemma cos_npi2:

      cos (pi * real n) = -1 ^ n

lemma sin_npi:

      sin (real n * pi) = 0

lemma sin_npi2:

      sin (pi * real n) = 0

lemma cos_two_pi:

      cos (2 * pi) = 1

lemma sin_two_pi:

      sin (2 * pi) = 0

lemma sin_gt_zero2:

      [| 0 < x; x < pi / 2 |] ==> 0 < sin x

lemma sin_less_zero:

      [| - pi / 2 < x; x < 0 |] ==> sin x < 0

lemma pi_less_4:

      pi < 4

lemma cos_gt_zero:

      [| 0 < x; x < pi / 2 |] ==> 0 < cos x

lemma cos_gt_zero_pi:

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

lemma cos_ge_zero:

      [| - (pi / 2) ≤ x; x ≤ pi / 2 |] ==> 0 ≤ cos x

lemma sin_gt_zero_pi:

      [| 0 < x; x < pi |] ==> 0 < sin x

lemma sin_ge_zero:

      [| 0 ≤ x; x ≤ pi |] ==> 0 ≤ sin x

lemma cos_total:

      [| -1 ≤ y; y ≤ 1 |] ==> ∃!x. 0 ≤ x ∧ x ≤ pi ∧ cos x = y

lemma sin_total:

      [| -1 ≤ y; y ≤ 1 |] ==> ∃!x. - (pi / 2) ≤ x ∧ x ≤ pi / 2 ∧ sin x = y

lemma reals_Archimedean4:

      [| 0 < y; 0 ≤ x |] ==> ∃n. real n * y ≤ x ∧ x < real (Suc n) * y

lemma cos_zero_lemma:

      [| 0 ≤ x; cos x = 0 |] ==> ∃n. odd n ∧ x = real n * (pi / 2)

lemma sin_zero_lemma:

      [| 0 ≤ x; sin x = 0 |] ==> ∃n. even n ∧ x = real n * (pi / 2)

lemma cos_zero_iff:

      (cos x = 0) =
      ((∃n. odd n ∧ x = real n * (pi / 2)) ∨ (∃n. odd n ∧ x = - (real n * (pi / 2))))

lemma sin_zero_iff:

      (sin x = 0) =
      ((∃n. even n ∧ x = real n * (pi / 2)) ∨
       (∃n. even n ∧ x = - (real n * (pi / 2))))

### Tangent

lemma tan_zero:

      tan 0 = 0

lemma tan_pi:

      tan pi = 0

lemma tan_npi:

      tan (real n * pi) = 0

lemma tan_minus:

      tan (- x) = - tan x

lemma tan_periodic:

      tan (x + 2 * pi) = tan x

lemma lemma_tan_add1:

      [| cos x ≠ 0; cos y ≠ 0 |] ==> 1 - tan x * tan y = cos (x + y) / (cos x * cos y)

lemma add_tan_eq:

      [| cos x ≠ 0; cos y ≠ 0 |] ==> tan x + tan y = sin (x + y) / (cos x * cos y)

lemma tan_add:

      [| cos x ≠ 0; cos y ≠ 0; cos (x + y) ≠ 0 |]
      ==> tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)

lemma tan_double:

      [| cos x ≠ 0; cos (2 * x) ≠ 0 |] ==> tan (2 * x) = 2 * tan x / (1 - (tan x)²)

lemma tan_gt_zero:

      [| 0 < x; x < pi / 2 |] ==> 0 < tan x

lemma tan_less_zero:

      [| - pi / 2 < x; x < 0 |] ==> tan x < 0

lemma lemma_DERIV_tan:

      cos x ≠ 0 ==> DERIV (%x. sin x / cos x) x :> inverse ((cos x)²)

lemma DERIV_tan:

      cos x ≠ 0 ==> DERIV tan x :> inverse ((cos x)²)

lemma LIM_cos_div_sin:

      (%x. cos x / sin x) -- pi / 2 --> 0

lemma lemma_tan_total:

      0 < y ==> ∃x>0. x < pi / 2 ∧ y < tan x

lemma tan_total_pos:

      0 ≤ y ==> ∃x≥0. x < pi / 2 ∧ tan x = y

lemma lemma_tan_total1:

      ∃x>- (pi / 2). x < pi / 2 ∧ tan x = y

lemma tan_total:

      ∃!x. - (pi / 2) < x ∧ x < pi / 2 ∧ tan x = y

lemma arcsin_pi:

      [| -1 ≤ y; y ≤ 1 |]
      ==> - (pi / 2) ≤ arcsin y ∧ arcsin y ≤ pi ∧ sin (arcsin y) = y

lemma arcsin:

      [| -1 ≤ y; y ≤ 1 |]
      ==> - (pi / 2) ≤ arcsin y ∧ arcsin y ≤ pi / 2 ∧ sin (arcsin y) = y

lemma sin_arcsin:

      [| -1 ≤ y; y ≤ 1 |] ==> sin (arcsin y) = y

lemma arcsin_bounded:

      [| -1 ≤ y; y ≤ 1 |] ==> - (pi / 2) ≤ arcsin y ∧ arcsin y ≤ pi / 2

lemma arcsin_lbound:

      [| -1 ≤ y; y ≤ 1 |] ==> - (pi / 2) ≤ arcsin y

lemma arcsin_ubound:

      [| -1 ≤ y; y ≤ 1 |] ==> arcsin y ≤ pi / 2

lemma arcsin_lt_bounded:

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

lemma arcsin_sin:

      [| - (pi / 2) ≤ x; x ≤ pi / 2 |] ==> arcsin (sin x) = x

lemma arcos:

      [| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arcos y ∧ arcos y ≤ pi ∧ cos (arcos y) = y

lemma cos_arcos:

      [| -1 ≤ y; y ≤ 1 |] ==> cos (arcos y) = y

lemma arcos_bounded:

      [| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arcos y ∧ arcos y ≤ pi

lemma arcos_lbound:

      [| -1 ≤ y; y ≤ 1 |] ==> 0 ≤ arcos y

lemma arcos_ubound:

      [| -1 ≤ y; y ≤ 1 |] ==> arcos y ≤ pi

lemma arcos_lt_bounded:

      [| -1 < y; y < 1 |] ==> 0 < arcos y ∧ arcos y < pi

lemma arcos_cos:

      [| 0 ≤ x; x ≤ pi |] ==> arcos (cos x) = x

lemma arcos_cos2:

      [| x ≤ 0; - pi ≤ x |] ==> arcos (cos x) = - x

lemma arctan:

      - (pi / 2) < arctan y ∧ arctan y < pi / 2 ∧ tan (arctan y) = y

lemma tan_arctan:

      tan (arctan y) = y

lemma arctan_bounded:

      - (pi / 2) < arctan y ∧ arctan y < pi / 2

lemma arctan_lbound:

      - (pi / 2) < arctan y

lemma arctan_ubound:

      arctan y < pi / 2

lemma arctan_tan:

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

lemma arctan_zero_zero:

      arctan 0 = 0

lemma cos_arctan_not_zero:

      cos (arctan x) ≠ 0

lemma tan_sec:

      cos x ≠ 0 ==> 1 + (tan x)² = (inverse (cos x))²

lemma

      sin (x + 1 / 2 * real (Suc m) * pi) = cos (x + 1 / 2 * real m * pi)

lemma

      sin (x + real (Suc m) * pi / 2) = cos (x + real m * pi / 2)

lemma DERIV_sin_add:

      DERIV (%x. sin (x + k)) xa :> cos (xa + k)

lemma sin_cos_npi:

      sin (real (Suc (2 * n)) * pi / 2) = -1 ^ n

lemma cos_2npi:

      cos (2 * real n * pi) = 1

lemma cos_3over2_pi:

      cos (3 / 2 * pi) = 0

lemma sin_2npi:

      sin (2 * real n * pi) = 0

lemma sin_3over2_pi:

      sin (3 / 2 * pi) = - 1

lemma

      cos (x + 1 / 2 * real (Suc m) * pi) = - sin (x + 1 / 2 * real m * pi)

lemma

      cos (x + real (Suc m) * pi / 2) = - sin (x + real m * pi / 2)

lemma cos_pi_eq_zero:

      cos (pi * real (Suc (2 * m)) / 2) = 0

lemma DERIV_cos_add:

      DERIV (%x. cos (x + k)) xa :> - sin (xa + k)

lemma isCont_cos:

      isCont cos x

lemma isCont_sin:

      isCont sin x

lemma isCont_exp:

      isCont exp x

lemma sin_zero_abs_cos_one:

      sin x = 0 ==> ¦cos x¦ = 1

lemma exp_eq_one_iff:

      (exp x = 1) = (x = 0)

lemma cos_one_sin_zero:

      cos x = 1 ==> sin x = 0

lemma real_root_less_mono:

      [| 0 ≤ x; x < y |] ==> root (Suc n) x < root (Suc n) y

lemma real_root_le_mono:

      [| 0 ≤ x; x ≤ y |] ==> root (Suc n) x ≤ root (Suc n) y

lemma real_root_less_iff:

      [| 0 ≤ x; 0 ≤ y |] ==> (root (Suc n) x < root (Suc n) y) = (x < y)

lemma real_root_le_iff:

      [| 0 ≤ x; 0 ≤ y |] ==> (root (Suc n) x ≤ root (Suc n) y) = (x ≤ y)

lemma real_root_eq_iff:

      [| 0 ≤ x; 0 ≤ y |] ==> (root (Suc n) x = root (Suc n) y) = (x = y)

lemma real_root_pos_unique:

      [| 0 ≤ x; 0 ≤ y; y ^ Suc n = x |] ==> root (Suc n) x = y

lemma real_root_mult:

      [| 0 ≤ x; 0 ≤ y |] ==> root (Suc n) (x * y) = root (Suc n) x * root (Suc n) y

lemma real_root_inverse:

      0 ≤ x ==> root (Suc n) (inverse x) = inverse (root (Suc n) x)

lemma real_root_divide:

      [| 0 ≤ x; 0 ≤ y |] ==> root (Suc n) (x / y) = root (Suc n) x / root (Suc n) y

lemma real_sqrt_less_mono:

      [| 0 ≤ x; x < y |] ==> sqrt x < sqrt y

lemma real_sqrt_le_mono:

      [| 0 ≤ x; x ≤ y |] ==> sqrt x ≤ sqrt y

lemma real_sqrt_less_iff:

      [| 0 ≤ x; 0 ≤ y |] ==> (sqrt x < sqrt y) = (x < y)

lemma real_sqrt_le_iff:

      [| 0 ≤ x; 0 ≤ y |] ==> (sqrt x ≤ sqrt y) = (x ≤ y)

lemma real_sqrt_eq_iff:

      [| 0 ≤ x; 0 ≤ y |] ==> (sqrt x = sqrt y) = (x = y)

lemma real_sqrt_sos_less_one_iff:

      (sqrt (x² + y²) < 1) = (x² + y² < 1)

lemma real_sqrt_sos_eq_one_iff:

      (sqrt (x² + y²) = 1) = (x² + y² = 1)

lemma real_divide_square_eq:

      r * a / (r * r) = a / r

### Theorems About Sqrt, Transcendental Functions for Complex

lemma le_real_sqrt_sumsq:

      x ≤ sqrt (x * x + y * y)

lemma minus_le_real_sqrt_sumsq:

      - x ≤ sqrt (x * x + y * y)

lemma lemma_real_divide_sqrt_ge_minus_one:

      0 < x ==> -1 ≤ x / sqrt (x * x + y * y)

lemma real_sqrt_sum_squares_gt_zero1:

      x < 0 ==> 0 < sqrt (x * x + y * y)

lemma real_sqrt_sum_squares_gt_zero2:

      0 < x ==> 0 < sqrt (x * x + y * y)

lemma real_sqrt_sum_squares_gt_zero3:

      x ≠ 0 ==> 0 < sqrt (x² + y²)

lemma real_sqrt_sum_squares_gt_zero3a:

      y ≠ 0 ==> 0 < sqrt (x² + y²)

lemma real_sqrt_sum_squares_eq_cancel:

      sqrt (x² + y²) = x ==> y = 0

lemma real_sqrt_sum_squares_eq_cancel2:

      sqrt (x² + y²) = y ==> x = 0

lemma lemma_real_divide_sqrt_le_one:

      x < 0 ==> x / sqrt (x * x + y * y) ≤ 1

lemma lemma_real_divide_sqrt_ge_minus_one2:

      x < 0 ==> -1 ≤ x / sqrt (x * x + y * y)

lemma lemma_real_divide_sqrt_le_one2:

      0 < x ==> x / sqrt (x * x + y * y) ≤ 1

lemma minus_sqrt_le:

      - sqrt (x * x + y * y) ≤ x

lemma minus_sqrt_le2:

      - sqrt (x * x + y * y) ≤ y

lemma not_neg_sqrt_sumsq:

      ¬ sqrt (x * x + y * y) < 0

lemma cos_x_y_ge_minus_one:

      -1 ≤ x / sqrt (x * x + y * y)

lemma cos_x_y_ge_minus_one1a:

      -1 ≤ y / sqrt (x * x + y * y)

lemma cos_x_y_le_one:

      x / sqrt (x * x + y * y) ≤ 1

lemma cos_x_y_le_one2:

      y / sqrt (x * x + y * y) ≤ 1

lemma cos_abs_x_y_ge_minus_one:

      -1 ≤ ¦x¦ / sqrt (x * x + y * y)

lemma cos_abs_x_y_le_one:

      ¦x¦ / sqrt (x * x + y * y) ≤ 1

lemma minus_pi_less_zero:

      - pi < 0

lemma arcos_ge_minus_pi:

      [| -1 ≤ y; y ≤ 1 |] ==> - pi ≤ arcos y

lemma lemma_divide_rearrange:

      [| x + y ≠ 0; 1 - z = x / (x + y) |] ==> z = y / (x + y)

lemma lemma_cos_sin_eq:

      [| 0 < x * x + y * y; 1 - (sin xa)² = (x / sqrt (x * x + y * y))² |]
      ==> (sin xa)² = (y / sqrt (x * x + y * y))²

lemma lemma_sin_cos_eq:

      [| 0 < x * x + y * y; 1 - (cos xa)² = (y / sqrt (x * x + y * y))² |]
      ==> (cos xa)² = (x / sqrt (x * x + y * y))²

lemma sin_x_y_disj:

      [| x ≠ 0; cos xa = x / sqrt (x * x + y * y) |]
      ==> sin xa = y / sqrt (x * x + y * y) ∨ sin xa = - y / sqrt (x * x + y * y)

lemma lemma_cos_not_eq_zero:

      x ≠ 0 ==> x / sqrt (x * x + y * y) ≠ 0

lemma cos_x_y_disj:

      [| x ≠ 0; sin xa = y / sqrt (x * x + y * y) |]
      ==> cos xa = x / sqrt (x * x + y * y) ∨ cos xa = - x / sqrt (x * x + y * y)

lemma real_sqrt_divide_less_zero:

      0 < y ==> - y / sqrt (x * x + y * y) < 0

lemma polar_ex1:

      [| x ≠ 0; 0 < y |] ==> ∃r a. x = r * cos a ∧ y = r * sin a

lemma real_sum_squares_cancel2a:

      x * x = - (y * y) ==> y = 0

lemma polar_ex2:

      [| x ≠ 0; y < 0 |] ==> ∃r a. x = r * cos a ∧ y = r * sin a

lemma polar_Ex:

      ∃r a. x = r * cos a ∧ y = r * sin a

lemma real_sqrt_ge_abs1:

      ¦x¦ ≤ sqrt (x² + y²)

lemma real_sqrt_ge_abs2:

      ¦y¦ ≤ sqrt (x² + y²)

lemma real_sqrt_two_gt_zero:

      0 < sqrt 2

lemma real_sqrt_two_ge_zero:

      0 ≤ sqrt 2

lemma real_sqrt_two_gt_one:

      1 < sqrt 2

lemma lemma_real_divide_sqrt_less:

      0 < u ==> u / sqrt 2 < u

lemma four_x_squared:

      4 * x² = (2 * x)²

lemma lemma_sqrt_hcomplex_capprox:

      [| 0 < u; x < u / 2; y < u / 2; 0 ≤ x; 0 ≤ y |] ==> sqrt (x² + y²) < u

### A Few Theorems Involving Ln, Derivatives, etc.

lemma lemma_DERIV_ln:

      DERIV ln z :> l ==> DERIV (%x. exp (ln x)) z :> exp (ln z) * l

lemma STAR_exp_ln:

      0 < z ==> (*f* (%x. exp (ln x))) z = z

lemma hypreal_add_Infinitesimal_gt_zero:

      [| e ∈ Infinitesimal; 0 < x |] ==> 0 < star_of x + e

lemma NSDERIV_exp_ln_one:

      0 < z ==> NSDERIV (%x. exp (ln x)) z :> 1

lemma DERIV_exp_ln_one:

      0 < z ==> DERIV (%x. exp (ln x)) z :> 1

lemma lemma_DERIV_ln2:

      [| 0 < z; DERIV ln z :> l |] ==> exp (ln z) * l = 1

lemma lemma_DERIV_ln3:

      [| 0 < z; DERIV ln z :> l |] ==> l = 1 / exp (ln z)

lemma lemma_DERIV_ln4:

      [| 0 < z; DERIV ln z :> l |] ==> l = 1 / z

lemma isCont_inv_fun:

      [| 0 < d; ∀z. ¦z - x¦ ≤ d --> g (f z) = z; ∀z. ¦z - x¦ ≤ d --> isCont f z |]
      ==> isCont g (f x)

lemma isCont_inv_fun_inv:

      [| 0 < d; ∀z. ¦z - x¦ ≤ d --> g (f z) = z; ∀z. ¦z - x¦ ≤ d --> isCont f z |]
      ==> ∃e>0. ∀y. 0 < ¦y - f x¦ ∧ ¦y - f x¦ < e --> f (g y) = y

lemma LIM_fun_gt_zero:

      [| f -- c --> l; 0 < l |] ==> ∃r>0. ∀x. x ≠ c ∧ ¦c - x¦ < r --> 0 < f x

lemma LIM_fun_less_zero:

      [| f -- c --> l; l < 0 |] ==> ∃r>0. ∀x. x ≠ c ∧ ¦c - x¦ < r --> f x < 0

lemma LIM_fun_not_zero:

      [| f -- c --> l; l ≠ 0 |] ==> ∃r>0. ∀x. x ≠ c ∧ ¦c - x¦ < r --> f x ≠ 0