# Theory MacLaurin

Up to index of Isabelle/HOL/HOL-Complex

theory MacLaurin
imports Log
begin

    (*  ID          : $Id: MacLaurin.thy,v 1.17 2005/07/27 09:28:19 paulson Exp $
        Author      : Jacques D. Fleuriot
        Copyright   : 2001 University of Edinburgh
        Conversion to Isar and new proofs by Lawrence C Paulson, 2004
    *)

    header{*MacLaurin Series*}

    theory MacLaurin
    imports Log
    begin

    subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}

    text{*This is a very long, messy proof even now that it's been broken down
    into lemmas.*}

    lemma Maclaurin_lemma:
        "0 < h ==>
         ∃B. f h = (∑m=0..<n. (j m / real (fact m)) * (h^m)) +
                   (B * ((h^n) / real(fact n)))"
    apply (rule_tac x = "(f h - (∑m=0..<n. (j m / real (fact m)) * h^m)) *
                     real(fact n) / (h^n)"
           in exI)
    apply (simp)
    done

    lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
    by arith

    text{*A crude tactic to differentiate by proof.*}
    ML
    {*
    exception DERIV_name;
    fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f
    |   get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f
    |   get_fun_name _ = raise DERIV_name;

    val deriv_rulesI = [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,
                        DERIV_Id,DERIV_const,DERIV_cos];

    val deriv_tac =
      SUBGOAL (fn (prem,i) =>
       (resolve_tac deriv_rulesI i) ORELSE
        ((rtac (read_instantiate [("f",get_fun_name prem)]
                         DERIV_chain2) i) handle DERIV_name => no_tac));;

    val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));
    *}

    lemma Maclaurin_lemma2:
          "[| ∀m t. m < n ∧ 0≤t ∧ t≤h --> DERIV (diff m) t :> diff (Suc m) t;
              n = Suc k;
            difg =
            (λm t. diff m t -
                   ((∑p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
                    B * (t ^ (n - m) / real (fact (n - m)))))|] ==>
            ∀m t. m < n & 0 ≤ t & t ≤ h -->
                        DERIV (difg m) t :> difg (Suc m) t"
    apply clarify
    apply (rule DERIV_diff)
    apply (simp (no_asm_simp))
    apply (tactic DERIV_tac)
    apply (tactic DERIV_tac)
    apply (rule_tac [2] lemma_DERIV_subst)
    apply (rule_tac [2] DERIV_quotient)
    apply (rule_tac [3] DERIV_const)
    apply (rule_tac [2] DERIV_pow)
      prefer 3 apply (simp add: fact_diff_Suc)
     prefer 2 apply simp
    apply (frule_tac m = m in less_add_one, clarify)
    apply (simp del: setsum_op_ivl_Suc)
    apply (insert sumr_offset4 [of 1])
    apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)
    apply (rule lemma_DERIV_subst)
    apply (rule DERIV_add)
    apply (rule_tac [2] DERIV_const)
    apply (rule DERIV_sumr, clarify)
     prefer 2 apply simp
    apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)
    apply (rule DERIV_cmult)
    apply (rule lemma_DERIV_subst)
    apply (best intro: DERIV_chain2 intro!: DERIV_intros)
    apply (subst fact_Suc)
    apply (subst real_of_nat_mult)
    apply (simp add: mult_ac)
    done

    lemma Maclaurin_lemma3:
         "[|∀k t. k < Suc m ∧ 0≤t & t≤h --> DERIV (difg k) t :> difg (Suc k) t;
            ∀k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0;  n < m; 0 < t;
            t < h|]
         ==> ∃ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0"
    apply (rule Rolle, assumption, simp)
    apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec)
    apply (rule DERIV_unique)
    prefer 2 apply assumption
    apply force
    apply (subgoal_tac "∀ta. 0 ≤ ta & ta ≤ t --> (difg (Suc n)) differentiable ta")
    apply (simp add: differentiable_def)
    apply (blast dest!: DERIV_isCont)
    apply (simp add: differentiable_def, clarify)
    apply (rule_tac x = "difg (Suc (Suc n)) ta" in exI)
    apply force
    apply (simp add: differentiable_def, clarify)
    apply (rule_tac x = "difg (Suc (Suc n)) x" in exI)
    apply force
    done

    lemma Maclaurin:
       "[| 0 < h; 0 < n; diff 0 = f;
           ∀m t. m < n & 0 ≤ t & t ≤ h --> DERIV (diff m) t :> diff (Suc m) t |]
        ==> ∃t. 0 < t &
                  t < h &
                  f h =
                  setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
                  (diff n t / real (fact n)) * h ^ n"
    apply (case_tac "n = 0", force)
    apply (drule not0_implies_Suc)
    apply (erule exE)
    apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma)
    apply (erule exE)
    apply (subgoal_tac "∃g.
         g = (%t. f t - (setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n} + (B * (t^n / real(fact n)))))")
     prefer 2 apply blast
    apply (erule exE)
    apply (subgoal_tac "g 0 = 0 & g h =0")
     prefer 2
     apply (simp del: setsum_op_ivl_Suc)
     apply (cut_tac n = m and k = 1 in sumr_offset2)
     apply (simp add: eq_diff_eq' del: setsum_op_ivl_Suc)
    apply (subgoal_tac "∃difg. difg = (%m t. diff m t - (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m} + (B * ((t ^ (n - m)) / real (fact (n - m))))))")
     prefer 2 apply blast
    apply (erule exE)
    apply (subgoal_tac "difg 0 = g")
     prefer 2 apply simp
    apply (frule Maclaurin_lemma2, assumption+)
    apply (subgoal_tac "∀ma. ma < n --> (∃t. 0 < t & t < h & difg (Suc ma) t = 0) ")
     apply (drule_tac x = m and P="%m. m<n --> (∃t. ?QQ m t)" in spec)
     apply (erule impE)
      apply (simp (no_asm_simp))
     apply (erule exE)
     apply (rule_tac x = t in exI)
     apply (simp del: realpow_Suc fact_Suc)
    apply (subgoal_tac "∀m. m < n --> difg m 0 = 0")
     prefer 2
     apply clarify
     apply simp
     apply (frule_tac m = ma in less_add_one, clarify)
     apply (simp del: setsum_op_ivl_Suc)
    apply (insert sumr_offset4 [of 1])
    apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)
    apply (subgoal_tac "∀m. m < n --> (∃t. 0 < t & t < h & DERIV (difg m) t :> 0) ")
    apply (rule allI, rule impI)
    apply (drule_tac x = ma and P="%m. m<n --> (∃t. ?QQ m t)" in spec)
    apply (erule impE, assumption)
    apply (erule exE)
    apply (rule_tac x = t in exI)
    (* do some tidying up *)
    apply (erule_tac [!] V= "difg = (%m t. diff m t - (setsum (%p. diff (m + p) 0 / real (fact p) * t ^ p) {0..<n-m} + B * (t ^ (n - m) / real (fact (n - m)))))"
           in thin_rl)
    apply (erule_tac [!] V="g = (%t. f t - (setsum (%m. diff m 0 / real (fact m) * t ^ m) {0..<n} + B * (t ^ n / real (fact n))))"
           in thin_rl)
    apply (erule_tac [!] V="f h = setsum (%m. diff m 0 / real (fact m) * h ^ m) {0..<n} + B * (h ^ n / real (fact n))"
           in thin_rl)
    (* back to business *)
    apply (simp (no_asm_simp))
    apply (rule DERIV_unique)
    prefer 2 apply blast
    apply force
    apply (rule allI, induct_tac "ma")
    apply (rule impI, rule Rolle, assumption, simp, simp)
    apply (subgoal_tac "∀t. 0 ≤ t & t ≤ h --> g differentiable t")
    apply (simp add: differentiable_def)
    apply (blast dest: DERIV_isCont)
    apply (simp add: differentiable_def, clarify)
    apply (rule_tac x = "difg (Suc 0) t" in exI)
    apply force
    apply (simp add: differentiable_def, clarify)
    apply (rule_tac x = "difg (Suc 0) x" in exI)
    apply force
    apply safe
    apply force
    apply (frule Maclaurin_lemma3, assumption+, safe)
    apply (rule_tac x = ta in exI, force)
    done

    lemma Maclaurin_objl:
         "0 < h & 0 < n & diff 0 = f &
           (∀m t. m < n & 0 ≤ t & t ≤ h --> DERIV (diff m) t :> diff (Suc m) t)
        --> (∃t. 0 < t &
                  t < h &
                  f h =
                  (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) +
                  diff n t / real (fact n) * h ^ n)"
    by (blast intro: Maclaurin)

    lemma Maclaurin2:
       "[| 0 < h; diff 0 = f;
           ∀m t.
              m < n & 0 ≤ t & t ≤ h --> DERIV (diff m) t :> diff (Suc m) t |]
        ==> ∃t. 0 < t &
                  t ≤ h &
                  f h =
                  (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) +
                  diff n t / real (fact n) * h ^ n"
    apply (case_tac "n", auto)
    apply (drule Maclaurin, auto)
    done

    lemma Maclaurin2_objl:
         "0 < h & diff 0 = f &
           (∀m t.
              m < n & 0 ≤ t & t ≤ h --> DERIV (diff m) t :> diff (Suc m) t)
        --> (∃t. 0 < t &
                  t ≤ h &
                  f h =
                  (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) +
                  diff n t / real (fact n) * h ^ n)"
    by (blast intro: Maclaurin2)

    lemma Maclaurin_minus:
       "[| h < 0; 0 < n; diff 0 = f;
           ∀m t. m < n & h ≤ t & t ≤ 0 --> DERIV (diff m) t :> diff (Suc m) t |]
        ==> ∃t. h < t &
                  t < 0 &
                  f h =
                  (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) +
                  diff n t / real (fact n) * h ^ n"
    apply (cut_tac f = "%x. f (-x)"
            and diff = "%n x. ((- 1) ^ n) * diff n (-x)"
            and h = "-h" and n = n in Maclaurin_objl)
    apply (simp)
    apply safe
    apply (subst minus_mult_right)
    apply (rule DERIV_cmult)
    apply (rule lemma_DERIV_subst)
    apply (rule DERIV_chain2 [where g=uminus])
    apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_Id)
    prefer 2 apply force
    apply force
    apply (rule_tac x = "-t" in exI, auto)
    apply (subgoal_tac "(∑m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
                        (∑m = 0..<n. diff m 0 * h ^ m / real(fact m))")
    apply (rule_tac [2] setsum_cong[OF refl])
    apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
    done

    lemma Maclaurin_minus_objl:
         "(h < 0 & 0 < n & diff 0 = f &
           (∀m t.
              m < n & h ≤ t & t ≤ 0 --> DERIV (diff m) t :> diff (Suc m) t))
        --> (∃t. h < t &
                  t < 0 &
                  f h =
                  (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) +
                  diff n t / real (fact n) * h ^ n)"
    by (blast intro: Maclaurin_minus)

    subsection{*More Convenient "Bidirectional" Version.*}

    (* not good for PVS sin_approx, cos_approx *)

    lemma Maclaurin_bi_le_lemma [rule_format]:
         "0 < n -->
           diff 0 0 =
           (∑m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
           diff n 0 * 0 ^ n / real (fact n)"
    by (induct "n", auto)

    lemma Maclaurin_bi_le:
       "[| diff 0 = f;
           ∀m t. m < n & abs t ≤ abs x --> DERIV (diff m) t :> diff (Suc m) t |]
        ==> ∃t. abs t ≤ abs x &
                  f x =
                  (∑m=0..<n. diff m 0 / real (fact m) * x ^ m) +
                  diff n t / real (fact n) * x ^ n"
    apply (case_tac "n = 0", force)
    apply (case_tac "x = 0")
    apply (rule_tac x = 0 in exI)
    apply (force simp add: Maclaurin_bi_le_lemma)
    apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
    txt{*Case 1, where @{term "x < 0"}*}
    apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
    apply (simp add: abs_if)
    apply (rule_tac x = t in exI)
    apply (simp add: abs_if)
    txt{*Case 2, where @{term "0 < x"}*}
    apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
    apply (simp add: abs_if)
    apply (rule_tac x = t in exI)
    apply (simp add: abs_if)
    done

    lemma Maclaurin_all_lt:
         "[| diff 0 = f;
             ∀m x. DERIV (diff m) x :> diff(Suc m) x;
            x ~= 0; 0 < n
          |] ==> ∃t. 0 < abs t & abs t < abs x &
                   f x = (∑m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
                         (diff n t / real (fact n)) * x ^ n"
    apply (rule_tac x = x and y = 0 in linorder_cases)
    prefer 2 apply blast
    apply (drule_tac [2] diff=diff in Maclaurin)
    apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
    apply (rule_tac [!] x = t in exI, auto)
    done

    lemma Maclaurin_all_lt_objl:
         "diff 0 = f &
          (∀m x. DERIV (diff m) x :> diff(Suc m) x) &
          x ~= 0 & 0 < n
          --> (∃t. 0 < abs t & abs t < abs x &
                   f x = (∑m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
                         (diff n t / real (fact n)) * x ^ n)"
    by (blast intro: Maclaurin_all_lt)

    lemma Maclaurin_zero [rule_format]:
         "x = (0::real)
          ==> 0 < n -->
              (∑m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
              diff 0 0"
    by (induct n, auto)

    lemma Maclaurin_all_le: "[| diff 0 = f;
            ∀m x. DERIV (diff m) x :> diff (Suc m) x
          |] ==> ∃t. abs t ≤ abs x &
                  f x = (∑m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
                        (diff n t / real (fact n)) * x ^ n"
    apply (insert linorder_le_less_linear [of n 0])
    apply (erule disjE, force)
    apply (case_tac "x = 0")
    apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
    apply (drule gr_implies_not0 [THEN not0_implies_Suc])
    apply (rule_tac x = 0 in exI, force)
    apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
    apply (rule_tac x = t in exI, auto)
    done

    lemma Maclaurin_all_le_objl: "diff 0 = f &
          (∀m x. DERIV (diff m) x :> diff (Suc m) x)
          --> (∃t. abs t ≤ abs x &
                  f x = (∑m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
                        (diff n t / real (fact n)) * x ^ n)"
    by (blast intro: Maclaurin_all_le)

    subsection{*Version for Exponential Function*}

    lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
          ==> (∃t. 0 < abs t &
                    abs t < abs x &
                    exp x = (∑m=0..<n. (x ^ m) / real (fact m)) +
                            (exp t / real (fact n)) * x ^ n)"
    by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)

    lemma Maclaurin_exp_le:
         "∃t. abs t ≤ abs x &
                exp x = (∑m=0..<n. (x ^ m) / real (fact m)) +
                           (exp t / real (fact n)) * x ^ n"
    by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)

    subsection{*Version for Sine Function*}

    lemma MVT2:
         "[| a < b; ∀x. a ≤ x & x ≤ b --> DERIV f x :> f'(x) |]
          ==> ∃z. a < z & z < b & (f b - f a = (b - a) * f'(z))"
    apply (drule MVT)
    apply (blast intro: DERIV_isCont)
    apply (force dest: order_less_imp_le simp add: differentiable_def)
    apply (blast dest: DERIV_unique order_less_imp_le)
    done

    lemma mod_exhaust_less_4:
         "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
    by (case_tac "m mod 4", auto, arith)

    lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
         "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
    by (induct "n", auto)

    lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
         "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
    by (induct "n", auto)

    lemma Suc_mult_two_diff_one [rule_format, simp]:
          "0 < n --> Suc (2 * n - 1) = 2*n"
    by (induct "n", auto)

    text{*It is unclear why so many variant results are needed.*}

    lemma Maclaurin_sin_expansion2:
         "∃t. abs t ≤ abs x &
           sin x =
           (∑m=0..<n. (if even m then 0
                           else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
                           x ^ m)
          + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
    apply (cut_tac f = sin and n = n and x = x
            and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
    apply safe
    apply (simp (no_asm))
    apply (simp (no_asm))
    apply (case_tac "n", clarify, simp, simp)
    apply (rule ccontr, simp)
    apply (drule_tac x = x in spec, simp)
    apply (erule ssubst)
    apply (rule_tac x = t in exI, simp)
    apply (rule setsum_cong[OF refl])
    apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
    done

    lemma Maclaurin_sin_expansion:
         "∃t. sin x =
           (∑m=0..<n. (if even m then 0
                           else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
                           x ^ m)
          + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
    apply (insert Maclaurin_sin_expansion2 [of x n])
    apply (blast intro: elim:);
    done

    lemma Maclaurin_sin_expansion3:
         "[| 0 < n; 0 < x |] ==>
           ∃t. 0 < t & t < x &
           sin x =
           (∑m=0..<n. (if even m then 0
                           else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
                           x ^ m)
          + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
    apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_objl)
    apply safe
    apply simp
    apply (simp (no_asm))
    apply (erule ssubst)
    apply (rule_tac x = t in exI, simp)
    apply (rule setsum_cong[OF refl])
    apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
    done

    lemma Maclaurin_sin_expansion4:
         "0 < x ==>
           ∃t. 0 < t & t ≤ x &
           sin x =
           (∑m=0..<n. (if even m then 0
                           else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
                           x ^ m)
          + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
    apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin2_objl)
    apply safe
    apply simp
    apply (simp (no_asm))
    apply (erule ssubst)
    apply (rule_tac x = t in exI, simp)
    apply (rule setsum_cong[OF refl])
    apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
    done

    subsection{*Maclaurin Expansion for Cosine Function*}

    lemma sumr_cos_zero_one [simp]:
     "(∑m=0..<(Suc n).
         (if even m then (- 1) ^ (m div 2)/(real  (fact m)) else 0) * 0 ^ m) = 1"
    by (induct "n", auto)

    lemma Maclaurin_cos_expansion:
         "∃t. abs t ≤ abs x &
           cos x =
           (∑m=0..<n. (if even m
                           then (- 1) ^ (m div 2)/(real (fact m))
                           else 0) *
                           x ^ m)
          + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
    apply (cut_tac f = cos and n = n and x = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
    apply safe
    apply (simp (no_asm))
    apply (simp (no_asm))
    apply (case_tac "n", simp)
    apply (simp del: setsum_op_ivl_Suc)
    apply (rule ccontr, simp)
    apply (drule_tac x = x in spec, simp)
    apply (erule ssubst)
    apply (rule_tac x = t in exI, simp)
    apply (rule setsum_cong[OF refl])
    apply (auto simp add: cos_zero_iff even_mult_two_ex)
    done

    lemma Maclaurin_cos_expansion2:
         "[| 0 < x; 0 < n |] ==>
           ∃t. 0 < t & t < x &
           cos x =
           (∑m=0..<n. (if even m
                           then (- 1) ^ (m div 2)/(real (fact m))
                           else 0) *
                           x ^ m)
          + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
    apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl)
    apply safe
    apply simp
    apply (simp (no_asm))
    apply (erule ssubst)
    apply (rule_tac x = t in exI, simp)
    apply (rule setsum_cong[OF refl])
    apply (auto simp add: cos_zero_iff even_mult_two_ex)
    done

    lemma Maclaurin_minus_cos_expansion:
         "[| x < 0; 0 < n |] ==>
           ∃t. x < t & t < 0 &
           cos x =
           (∑m=0..<n. (if even m
                           then (- 1) ^ (m div 2)/(real (fact m))
                           else 0) *
                           x ^ m)
          + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
    apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_minus_objl)
    apply safe
    apply simp
    apply (simp (no_asm))
    apply (erule ssubst)
    apply (rule_tac x = t in exI, simp)
    apply (rule setsum_cong[OF refl])
    apply (auto simp add: cos_zero_iff even_mult_two_ex)
    done

    (* ------------------------------------------------------------------------- *)
    (* Version for ln(1 +/- x). Where is it??                                    *)
    (* ------------------------------------------------------------------------- *)

    lemma sin_bound_lemma:
        "[|x = y; abs u ≤ (v::real) |] ==> ¦(x + u) - y¦ ≤ v"
    by auto

    lemma Maclaurin_sin_bound:
      "abs(sin x - (∑m=0..<n. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
      x ^ m))  ≤ inverse(real (fact n)) * ¦x¦ ^ n"
    proof -
      have "!! x (y::real). x ≤ 1 ==> 0 ≤ y ==> x * y ≤ 1 * y"
        by (rule_tac mult_right_mono,simp_all)
      note est = this[simplified]
      show ?thesis
        apply (cut_tac f=sin and n=n and x=x and
          diff = "%n x. if n mod 4 = 0 then sin(x) else if n mod 4 = 1 then cos(x) else if n mod 4 = 2 then -sin(x) else -cos(x)"
          in Maclaurin_all_le_objl)
        apply safe
        apply simp
        apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
        apply (cut_tac m=m in mod_exhaust_less_4, safe, simp+)
        apply (rule DERIV_minus, simp+)
        apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
        apply (erule ssubst)
        apply (rule sin_bound_lemma)
        apply (rule setsum_cong[OF refl])
        apply (rule_tac f = "%u. u * (x^xa)" in arg_cong)
        apply (subst even_even_mod_4_iff)
        apply (cut_tac m=xa in mod_exhaust_less_4, simp, safe)
        apply (simp_all add:even_num_iff)
        apply (drule lemma_even_mod_4_div_2[simplified])
        apply(simp add: numeral_2_eq_2 divide_inverse)
        apply (drule lemma_odd_mod_4_div_2)
        apply (simp add: numeral_2_eq_2 divide_inverse)
        apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
                       simp add: est mult_nonneg_nonneg mult_ac divide_inverse
                              power_abs [symmetric] abs_mult)
        done
    qed

    end

### Maclaurin's Theorem with Lagrange Form of Remainder

lemma Maclaurin_lemma:

      0 < h
      ==> ∃B. f h =
              (∑m = 0..<n. j m / real (fact m) * h ^ m) + B * (h ^ n / real (fact n))

lemma eq_diff_eq':

      (x = y - z) = (y = x + z)

lemma Maclaurin_lemma2:

      [| ∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t;
         n = Suc k;
         difg =
         (%m t. diff m t -
                ((∑p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
                 B * (t ^ (n - m) / real (fact (n - m))))) |]
      ==> ∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (difg m) t :> difg (Suc m) t

lemma Maclaurin_lemma3:

      [| ∀k t. k < Suc m ∧ 0 ≤ t ∧ t ≤ h --> DERIV (difg k) t :> difg (Suc k) t;
         ∀k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0; n < m; 0 < t; t < h |]
      ==> ∃ta>0. ta < t ∧ DERIV (difg (Suc n)) ta :> 0

lemma Maclaurin:

      [| 0 < h; 0 < n; diff 0 = f;
         ∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t |]
      ==> ∃t>0. t < h ∧
                f h =
                (∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
                diff n t / real (fact n) * h ^ n

lemma Maclaurin_objl:

      0 < h ∧
      0 < n ∧
      diff 0 = f ∧
      (∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t) -->
      (∃t>0. t < h ∧
             f h =
             (∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
             diff n t / real (fact n) * h ^ n)

lemma Maclaurin2:

      [| 0 < h; diff 0 = f;
         ∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t |]
      ==> ∃t>0. t ≤ h ∧
                f h =
                (∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
                diff n t / real (fact n) * h ^ n

lemma Maclaurin2_objl:

      0 < h ∧
      diff 0 = f ∧
      (∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t) -->
      (∃t>0. t ≤ h ∧
             f h =
             (∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
             diff n t / real (fact n) * h ^ n)

lemma Maclaurin_minus:

      [| h < 0; 0 < n; diff 0 = f;
         ∀m t. m < n ∧ h ≤ t ∧ t ≤ 0 --> DERIV (diff m) t :> diff (Suc m) t |]
      ==> ∃t>h. t < 0 ∧
                f h =
                (∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
                diff n t / real (fact n) * h ^ n

lemma Maclaurin_minus_objl:

      h < 0 ∧
      0 < n ∧
      diff 0 = f ∧
      (∀m t. m < n ∧ h ≤ t ∧ t ≤ 0 --> DERIV (diff m) t :> diff (Suc m) t) -->
      (∃t>h. t < 0 ∧
             f h =
             (∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
             diff n t / real (fact n) * h ^ n)

### More Convenient "Bidirectional" Version.

lemma Maclaurin_bi_le_lemma:

      0 < n
      ==> diff 0 (0::'a) =
          (∑m = 0..<n. diff m (0::'a) * 0 ^ m / real (fact m)) +
          diff n (0::'a) * 0 ^ n / real (fact n)

lemma Maclaurin_bi_le:

      [| diff 0 = f; ∀m t. m < n ∧ ¦t¦ ≤ ¦x¦ --> DERIV (diff m) t :> diff (Suc m) t |]
      ==> ∃t. ¦t¦ ≤ ¦x¦ ∧
              f x =
              (∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
              diff n t / real (fact n) * x ^ n

lemma Maclaurin_all_lt:

      [| diff 0 = f; ∀m x. DERIV (diff m) x :> diff (Suc m) x; x ≠ 0; 0 < n |]
      ==> ∃t. 0 < ¦t¦ ∧
              ¦t¦ < ¦x¦ ∧
              f x =
              (∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
              diff n t / real (fact n) * x ^ n

lemma Maclaurin_all_lt_objl:

      diff 0 = f ∧ (∀m x. DERIV (diff m) x :> diff (Suc m) x) ∧ x ≠ 0 ∧ 0 < n -->
      (∃t. 0 < ¦t¦ ∧
           ¦t¦ < ¦x¦ ∧
           f x =
           (∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
           diff n t / real (fact n) * x ^ n)

lemma Maclaurin_zero:

      [| x = 0; 0 < n |] ==> (∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0

lemma Maclaurin_all_le:

      [| diff 0 = f; ∀m x. DERIV (diff m) x :> diff (Suc m) x |]
      ==> ∃t. ¦t¦ ≤ ¦x¦ ∧
              f x =
              (∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
              diff n t / real (fact n) * x ^ n

lemma Maclaurin_all_le_objl:

      diff 0 = f ∧ (∀m x. DERIV (diff m) x :> diff (Suc m) x) -->
      (∃t. ¦t¦ ≤ ¦x¦ ∧
           f x =
           (∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
           diff n t / real (fact n) * x ^ n)

### Version for Exponential Function

lemma Maclaurin_exp_lt:

      [| x ≠ 0; 0 < n |]
      ==> ∃t. 0 < ¦t¦ ∧
              ¦t¦ < ¦x¦ ∧
              exp x =
              (∑m = 0..<n. x ^ m / real (fact m)) + exp t / real (fact n) * x ^ n

lemma Maclaurin_exp_le:

      ∃t. ¦t¦ ≤ ¦x¦ ∧
          exp x = (∑m = 0..<n. x ^ m / real (fact m)) + exp t / real (fact n) * x ^ n

### Version for Sine Function

lemma MVT2:

      [| a < b; ∀x. a ≤ x ∧ x ≤ b --> DERIV f x :> f' x |]
      ==> ∃z>a. z < b ∧ f b - f a = (b - a) * f' z

lemma mod_exhaust_less_4:

      m mod 4 = 0 ∨ m mod 4 = 1 ∨ m mod 4 = 2 ∨ m mod 4 = 3

lemma Suc_Suc_mult_two_diff_two:

      0 < n ==> Suc (Suc (2 * n - 2)) = 2 * n

lemma lemma_Suc_Suc_4n_diff_2:

      0 < n ==> Suc (Suc (4 * n - 2)) = 4 * n

lemma Suc_mult_two_diff_one:

      0 < n ==> Suc (2 * n - 1) = 2 * n

lemma Maclaurin_sin_expansion2:

      ∃t. ¦t¦ ≤ ¦x¦ ∧
          sin x =
          (∑m = 0..<n.
              (if even m then 0 else (- 1) ^ ((m - Suc 0) div 2) / real (fact m)) *
              x ^ m) +
          sin (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n

lemma Maclaurin_sin_expansion:

      ∃t. sin x =
          (∑m = 0..<n.
              (if even m then 0 else (- 1) ^ ((m - Suc 0) div 2) / real (fact m)) *
              x ^ m) +
          sin (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n

lemma Maclaurin_sin_expansion3:

      [| 0 < n; 0 < x |]
      ==> ∃t>0. t < x ∧
                sin x =
                (∑m = 0..<n.
                    (if even m then 0
                     else (- 1) ^ ((m - Suc 0) div 2) / real (fact m)) *
                    x ^ m) +
                sin (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n

lemma Maclaurin_sin_expansion4:

      0 < x
      ==> ∃t>0. t ≤ x ∧
                sin x =
                (∑m = 0..<n.
                    (if even m then 0
                     else (- 1) ^ ((m - Suc 0) div 2) / real (fact m)) *
                    x ^ m) +
                sin (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n

### Maclaurin Expansion for Cosine Function

lemma sumr_cos_zero_one:

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

lemma Maclaurin_cos_expansion:

      ∃t. ¦t¦ ≤ ¦x¦ ∧
          cos x =
          (∑m = 0..<n.
              (if even m then (- 1) ^ (m div 2) / real (fact m) else 0) * x ^ m) +
          cos (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n

lemma Maclaurin_cos_expansion2:

      [| 0 < x; 0 < n |]
      ==> ∃t>0. t < x ∧
                cos x =
                (∑m = 0..<n.
                    (if even m then (- 1) ^ (m div 2) / real (fact m) else 0) *
                    x ^ m) +
                cos (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n

lemma Maclaurin_minus_cos_expansion:

      [| x < 0; 0 < n |]
      ==> ∃t>x. t < 0 ∧
                cos x =
                (∑m = 0..<n.
                    (if even m then (- 1) ^ (m div 2) / real (fact m) else 0) *
                    x ^ m) +
                cos (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n

lemma sin_bound_lemma:

      [| x = y; ¦u¦ ≤ v |] ==> ¦x + u - y¦ ≤ v

lemma Maclaurin_sin_bound:

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