(*  Title       : MacLaurin.thy

    Author      : Jacques D. Fleuriot

    Copyright   : 2001 University of Edinburgh

    Description : MacLaurin series

*)

Goal "sumr 0 n (%m. f (m + k)) = sumr 0 (n + k) f - sumr 0 k f";

by (induct_tac "n" 1);

by Auto_tac;

qed "sumr_offset";

Goal "ALL f. sumr 0 n (%m. f (m + k)) = sumr 0 (n + k) f - sumr 0 k f";

by (induct_tac "n" 1);

by Auto_tac;

qed "sumr_offset2";

Goal "sumr 0 (n + k) f = sumr 0 n (%m. f (m + k)) + sumr 0 k f";

by (simp_tac (simpset() addsimps [sumr_offset]) 1);

qed "sumr_offset3";

Goal "ALL n f. sumr 0 (n + k) f = sumr 0 n (%m. f (m + k)) + sumr 0 k f";

by (simp_tac (simpset() addsimps [sumr_offset]) 1);

qed "sumr_offset4";

Goal "0 < n ==> \

\     sumr (Suc 0) (Suc n) (%n. (if even(n) then 0 else \

\            ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n) = \

\     sumr 0 (Suc n) (%n. (if even(n) then 0 else \

\            ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n)";

by (res_inst_tac [("n1","1")] (sumr_split_add RS subst) 1);

by Auto_tac;

qed "sumr_from_1_from_0";

(*---------------------------------------------------------------------------*)

(* Maclaurin's theorem with Lagrange form of remainder                       *)

(*---------------------------------------------------------------------------*)

(* Annoying: Proof is now even longer due mostly to 

   change in behaviour of simplifier  since Isabelle99 *)

Goal " [| 0 < h; 0 < n; diff 0 = f; \

\      ALL m t. \

\         m < n & 0 <= t & t <= h --> DERIV (diff m) t :> diff (Suc m) t |] \

\   ==> EX t. 0 < t & \

\             t < h & \

\             f h = \

\             sumr 0 n (%m. (diff m 0 / real (fact m)) * h ^ m) + \

\             (diff n t / real (fact n)) * h ^ n";

by (case_tac "n = 0" 1);

by (Force_tac 1);

by (dtac not0_implies_Suc 1);

by (etac exE 1);

by (subgoal_tac 

     "EX B. f h = sumr 0 n (%m. (diff m 0 / real (fact m)) * (h ^ m)) \

\                  + (B * ((h ^ n) / real (fact n)))" 1);

by (simp_tac (HOL_ss addsimps [real_add_commute, real_divide_def,

    ARITH_PROVE "(x = z + (y::real)) = (x - y = z)"]) 2);

by (res_inst_tac 

  [("x","(f(h) - sumr 0 n (%m. (diff(m)(0) / real (fact m)) * (h ^ m))) \

\        * real (fact n) / (h ^ n)")] exI 2);

by (simp_tac (HOL_ss addsimps [real_mult_assoc,real_divide_def]) 2);

 by (rtac (CLAIM "x = (1::real) ==>  a = a * (x::real)") 2);

by (asm_simp_tac (HOL_ss addsimps 

    [CLAIM "(a::real) * (b * (c * d)) = (d * a) * (b * c)"]

     delsimps [realpow_Suc]) 2);

by (stac left_inverse 2);

by (stac left_inverse 3);

by (rtac (real_not_refl2 RS not_sym) 2);

by (etac zero_less_power 2);

by (rtac real_of_nat_fact_not_zero 2);

by (Simp_tac 2);

by (etac exE 1);

by (cut_inst_tac [("b","%t. f t - \

\      (sumr 0 n (%m. (diff m 0 / real (fact m)) * (t ^ m)) + \

\                       (B * ((t ^ n) / real (fact n))))")] 

    (CLAIM "EX g. g = b") 1);

by (etac exE 1);

by (subgoal_tac "g 0 = 0 & g h =0" 1);

by (asm_simp_tac (simpset() addsimps 

    [ARITH_PROVE "(x - y = z) = (x = z + (y::real))"]

    delsimps [sumr_Suc]) 2);

by (cut_inst_tac [("n","m"),("k","1")] sumr_offset2 2);

by (asm_full_simp_tac (simpset() addsimps 

    [ARITH_PROVE "(x = y - z) = (y = x + (z::real))"]

    delsimps [sumr_Suc]) 2);

by (cut_inst_tac [("b","%m t. diff m t - \

\      (sumr 0 (n - m) (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) \

\       + (B * ((t ^ (n - m)) / real (fact(n - m)))))")] 

    (CLAIM "EX difg. difg = b") 1);

by (etac exE 1);

by (subgoal_tac "difg 0 = g" 1);

by (asm_simp_tac (simpset() delsimps [realpow_Suc,fact_Suc]) 2);

by (subgoal_tac "ALL m t. m < n & 0 <= t & t <= h --> \

\                   DERIV (difg m) t :> difg (Suc m) t" 1);

by (Clarify_tac 2);

by (rtac DERIV_diff 2);

by (Asm_simp_tac 2);

by DERIV_tac;

by DERIV_tac;

by (rtac lemma_DERIV_subst 3);

by (rtac DERIV_quotient 3);

by (rtac DERIV_const 4);

by (rtac DERIV_pow 3);

by (asm_simp_tac (simpset() addsimps [inverse_mult_distrib,

    CLAIM_SIMP "(a::real) * b * c * (d * e) = a * b * (c * d) * e" 

    mult_ac,fact_diff_Suc]) 4);

by (Asm_simp_tac 3);

by (forw_inst_tac [("m","ma")] less_add_one 2);

by (Clarify_tac 2);

by (asm_simp_tac (simpset() addsimps 

    [CLAIM "Suc m = ma + d + 1 ==> m - ma = d"]

    delsimps [sumr_Suc]) 2);

by (asm_simp_tac (simpset() addsimps [(simplify (simpset() delsimps [sumr_Suc]) 

          (read_instantiate [("k","1")] sumr_offset4))] 

    delsimps [sumr_Suc,fact_Suc,realpow_Suc]) 2);

by (rtac lemma_DERIV_subst 2);

by (rtac DERIV_add 2);

by (rtac DERIV_const 3);

by (rtac DERIV_sumr 2);

by (Clarify_tac 2);

by (Simp_tac 3);

by (simp_tac (simpset() addsimps [real_divide_def,real_mult_assoc] 

    delsimps [fact_Suc,realpow_Suc]) 2);

by (rtac DERIV_cmult 2);

by (rtac lemma_DERIV_subst 2);

by DERIV_tac;

by (stac fact_Suc 2);

by (stac real_of_nat_mult 2);

by (simp_tac (simpset() addsimps [inverse_mult_distrib] @

    mult_ac) 2);

by (subgoal_tac "ALL ma. ma < n --> \

\        (EX t. 0 < t & t < h & difg (Suc ma) t = 0)" 1);

by (rotate_tac 11 1);

by (dres_inst_tac [("x","m")] spec 1);

by (etac impE 1);

by (Asm_simp_tac 1);

by (etac exE 1);

by (res_inst_tac [("x","t")] exI 1);

by (asm_full_simp_tac (simpset() addsimps 

     [ARITH_PROVE "(x - y = 0) = (y = (x::real))"] 

      delsimps [realpow_Suc,fact_Suc]) 1);

by (subgoal_tac "ALL m. m < n --> difg m 0 = 0" 1);

by (Clarify_tac 2);

by (Asm_simp_tac 2);

by (forw_inst_tac [("m","ma")] less_add_one 2);

by (Clarify_tac 2);

by (asm_simp_tac (simpset() delsimps [sumr_Suc]) 2);

by (asm_simp_tac (simpset() addsimps [(simplify (simpset() delsimps [sumr_Suc]) 

          (read_instantiate [("k","1")] sumr_offset4))] 

    delsimps [sumr_Suc,fact_Suc,realpow_Suc]) 2);

by (subgoal_tac "ALL m. m < n --> (EX t. 0 < t & t < h & \

\                DERIV (difg m) t :> 0)" 1);

by (rtac allI 1 THEN rtac impI 1);

by (rotate_tac 12 1);

by (dres_inst_tac [("x","ma")] spec 1);

by (etac impE 1 THEN assume_tac 1);

by (etac exE 1);

by (res_inst_tac [("x","t")] exI 1);

(* do some tidying up *)

by (ALLGOALS(thin_tac "difg = \

\          (%m t. diff m t - \

\                 (sumr 0 (n - m) \

\                   (%p. diff (m + p) 0 / real (fact p) * t ^ p) + \

\                  B * (t ^ (n - m) / real (fact (n - m)))))"));

by (ALLGOALS(thin_tac "g = \

\          (%t. f t - \

\               (sumr 0 n (%m. diff m 0 / real  (fact m) * t ^ m) + \

\                B * (t ^ n / real (fact n))))"));

by (ALLGOALS(thin_tac "f h = \

\          sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \

\          B * (h ^ n / real (fact n))"));

(* back to business *)

by (Asm_simp_tac 1);

by (rtac DERIV_unique 1);

by (Blast_tac 2);

by (Force_tac 1);

by (rtac allI 1 THEN induct_tac "ma" 1);

by (rtac impI 1 THEN rtac Rolle 1);

by (assume_tac 1);

by (Asm_full_simp_tac 1);

by (Asm_full_simp_tac 1);

by (subgoal_tac "ALL t. 0 <= t & t <= h --> g differentiable t" 1);

by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 1);

by (blast_tac (claset() addDs [DERIV_isCont]) 1);

by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 1);

by (Clarify_tac 1);

by (res_inst_tac [("x","difg (Suc 0) t")] exI 1);

by (Force_tac 1);

by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 1);

by (Clarify_tac 1);

by (res_inst_tac [("x","difg (Suc 0) x")] exI 1);

by (Force_tac 1);

by (Step_tac 1);

by (Force_tac 1);

by (subgoal_tac "EX ta. 0 < ta & ta < t & \

\                DERIV difg (Suc n) ta :> 0" 1);

by (rtac Rolle 2 THEN assume_tac 2);

by (Asm_full_simp_tac 2);

by (rotate_tac 2 2);

by (dres_inst_tac [("x","n")] spec 2);

by (ftac (ARITH_PROVE "n < m  ==> n < Suc m") 2);

by (rtac DERIV_unique 2);

by (assume_tac 3);

by (Force_tac 2);

by (subgoal_tac 

    "ALL ta. 0 <= ta & ta <= t --> (difg (Suc n)) differentiable ta" 2);

by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 2);

by (blast_tac (claset() addSDs [DERIV_isCont]) 2);

by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 2);

by (Clarify_tac 2);

by (res_inst_tac [("x","difg (Suc (Suc n)) ta")] exI 2);

by (Force_tac 2);

by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 2);

by (Clarify_tac 2);

by (res_inst_tac [("x","difg (Suc (Suc n)) x")] exI 2);

by (Force_tac 2);

by (Step_tac 1);

by (res_inst_tac [("x","ta")] exI 1);

by (Force_tac 1);

qed "Maclaurin";

Goal "0 < h & 0 < n & diff 0 = f & \

\      (ALL m t. \

\         m < n & 0 <= t & t <= h --> DERIV (diff m) t :> diff (Suc m) t) \

\   --> (EX t. 0 < t & \

\             t < h & \

\             f h = \

\             sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \

\             diff n t / real (fact n) * h ^ n)";

by (blast_tac (claset() addIs [Maclaurin]) 1);

qed "Maclaurin_objl";

Goal " [| 0 < h; diff 0 = f; \

\      ALL m t. \

\         m < n & 0 <= t & t <= h --> DERIV (diff m) t :> diff (Suc m) t |] \

\   ==> EX t. 0 < t & \

\             t <= h & \

\             f h = \

\             sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \

\             diff n t / real (fact n) * h ^ n";

by (case_tac "n" 1);

by Auto_tac;

by (dtac Maclaurin 1 THEN Auto_tac);

qed "Maclaurin2";

Goal "0 < h & diff 0 = f & \

\      (ALL m t. \

\         m < n & 0 <= t & t <= h --> DERIV (diff m) t :> diff (Suc m) t) \

\   --> (EX t. 0 < t & \

\             t <= h & \

\             f h = \

\             sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \

\             diff n t / real (fact n) * h ^ n)";

by (blast_tac (claset() addIs [Maclaurin2]) 1);

qed "Maclaurin2_objl";

Goal " [| h < 0; 0 < n; diff 0 = f; \

\      ALL m t. \

\         m < n & h <= t & t <= 0 --> DERIV (diff m) t :> diff (Suc m) t |] \

\   ==> EX t. h < t & \

\             t < 0 & \

\             f h = \

\             sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \

\             diff n t / real (fact n) * h ^ n";

by (cut_inst_tac [("f","%x. f (-x)"),

                 ("diff","%n x. ((- 1) ^ n) * diff n (-x)"),

                 ("h","-h"),("n","n")] Maclaurin_objl 1);

by (Asm_full_simp_tac 1);

by (etac impE 1 THEN Step_tac 1);

by (stac minus_mult_right 1);

by (rtac DERIV_cmult 1);

by (rtac lemma_DERIV_subst 1);

by (rtac (read_instantiate [("g","uminus")] DERIV_chain2) 1);

by (rtac DERIV_minus 2 THEN rtac DERIV_Id 2);

by (Force_tac 2);

by (Force_tac 1);

by (res_inst_tac [("x","-t")] exI 1);

by Auto_tac;

by (rtac (CLAIM "[| x = x'; y = y' |] ==> x + y = x' + (y'::real)") 1);

by (rtac sumr_fun_eq 1);

by (Asm_full_simp_tac 1);

by (auto_tac (claset(),simpset() addsimps [real_divide_def,

    CLAIM "((a * b) * c) * d = (b * c) * (a * (d::real))",

    power_mult_distrib RS sym]));

qed "Maclaurin_minus";

Goal "(h < 0 & 0 < n & diff 0 = f & \

\      (ALL m t. \

\         m < n & h <= t & t <= 0 --> DERIV (diff m) t :> diff (Suc m) t))\

\   --> (EX t. h < t & \

\             t < 0 & \

\             f h = \

\             sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \

\             diff n t / real (fact n) * h ^ n)";

by (blast_tac (claset() addIs [Maclaurin_minus]) 1);

qed "Maclaurin_minus_objl";

(* ------------------------------------------------------------------------- *)

(* More convenient "bidirectional" version.                                  *)

(* ------------------------------------------------------------------------- *)

(* not good for PVS sin_approx, cos_approx *)

Goal " [| diff 0 = f; \

\      ALL m t. \

\         m < n & abs t <= abs x --> DERIV (diff m) t :> diff (Suc m) t |] \

\   ==> EX t. abs t <= abs x & \

\             f x = \

\             sumr 0 n (%m. diff m 0 / real (fact m) * x ^ m) + \

\             diff n t / real (fact n) * x ^ n";

by (case_tac "n = 0" 1);

by (Force_tac 1);

by (case_tac "x = 0" 1);

by (res_inst_tac [("x","0")] exI 1);

by (Asm_full_simp_tac 1);

by (res_inst_tac [("P","0 < n")] impE 1);

by (assume_tac 2 THEN assume_tac 2);

by (induct_tac "n" 1);

by (Simp_tac 1);

by Auto_tac;

by (cut_inst_tac [("x","x"),("y","0")] linorder_less_linear 1);

by Auto_tac;

by (cut_inst_tac [("f","diff 0"),

                 ("diff","diff"),

                 ("h","x"),("n","n")] Maclaurin_objl 2);

by (Step_tac 2);

by (blast_tac (claset() addDs 

    [ARITH_PROVE "[|(0::real) <= t;t <= x |] ==> abs t <= abs x"]) 2);

by (res_inst_tac [("x","t")] exI 2);

by (force_tac (claset() addIs 

    [ARITH_PROVE "[| 0 < t; (t::real) < x|] ==> abs t <= abs x"],simpset()) 2);

by (cut_inst_tac [("f","diff 0"),

                 ("diff","diff"),

                 ("h","x"),("n","n")] Maclaurin_minus_objl 1);

by (Step_tac 1);

by (blast_tac (claset() addDs 

    [ARITH_PROVE "[|x <= t;t <= (0::real) |] ==> abs t <= abs x"]) 1);

by (res_inst_tac [("x","t")] exI 1);

by (force_tac (claset() addIs 

    [ARITH_PROVE "[| x < t; (t::real) < 0|] ==> abs t <= abs x"],simpset()) 1);

qed "Maclaurin_bi_le";

Goal "[| diff 0 = f; \

\        ALL m x. DERIV (diff m) x :> diff(Suc m) x; \ 

\       x ~= 0; 0 < n \

\     |] ==> EX t. 0 < abs t & abs t < abs x & \

\              f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) + \

\                    (diff n t / real (fact n)) * x ^ n";

by (res_inst_tac [("x","x"),("y","0")] linorder_cases 1);

by (Blast_tac 2);

by (dtac Maclaurin_minus 1);

by (dtac Maclaurin 5);

by (TRYALL(assume_tac));

by (Blast_tac 1);

by (Blast_tac 2);

by (Step_tac 1);

by (ALLGOALS(res_inst_tac [("x","t")] exI));

by (Step_tac 1);

by (ALLGOALS(arith_tac));

qed "Maclaurin_all_lt";

Goal "diff 0 = f & \

\     (ALL m x. DERIV (diff m) x :> diff(Suc m) x) & \

\     x ~= 0 & 0 < n \

\     --> (EX t. 0 < abs t & abs t < abs x & \

\              f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) + \

\                    (diff n t / real (fact n)) * x ^ n)";

by (blast_tac (claset() addIs [Maclaurin_all_lt]) 1);

qed "Maclaurin_all_lt_objl";

Goal "x = (0::real)  \

\     ==> 0 < n --> \

\         sumr 0 n (%m. (diff m (0::real) / real (fact m)) * x ^ m) = \

\         diff 0 0";

by (Asm_simp_tac 1);

by (induct_tac "n" 1);

by Auto_tac; 

qed_spec_mp "Maclaurin_zero";

Goal "[| diff 0 = f; \

\       ALL m x. DERIV (diff m) x :> diff (Suc m) x \

\     |] ==> EX t. abs t <= abs x & \

\             f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) + \

\                   (diff n t / real (fact n)) * x ^ n";

by (cut_inst_tac [("n","n"),("m","0")] 

       (ARITH_PROVE "n <= m | m < (n::nat)") 1);

by (etac disjE 1);

by (Force_tac 1);

by (case_tac "x = 0" 1);

by (forw_inst_tac [("diff","diff"),("n","n")] Maclaurin_zero 1);

by (assume_tac 1);

by (dtac (gr_implies_not0 RS  not0_implies_Suc) 1);

by (res_inst_tac [("x","0")] exI 1);

by (Force_tac 1);

by (forw_inst_tac [("diff","diff"),("n","n")] Maclaurin_all_lt 1);

by (TRYALL(assume_tac));

by (Step_tac 1);

by (res_inst_tac [("x","t")] exI 1);

by Auto_tac;

qed "Maclaurin_all_le";

Goal "diff 0 = f & \

\     (ALL m x. DERIV (diff m) x :> diff (Suc m) x)  \

\     --> (EX t. abs t <= abs x & \

\             f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) + \

\                   (diff n t / real (fact n)) * x ^ n)";

by (blast_tac (claset() addIs [Maclaurin_all_le]) 1);

qed "Maclaurin_all_le_objl";

(* ------------------------------------------------------------------------- *)

(* Version for exp.                                                          *)

(* ------------------------------------------------------------------------- *)

Goal "[| x ~= 0; 0 < n |] \

\     ==> (EX t. 0 < abs t & \

\               abs t < abs x & \

\               exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) + \

\                       (exp t / real (fact n)) * x ^ n)";

by (cut_inst_tac [("diff","%n. exp"),("f","exp"),("x","x"),("n","n")] 

    Maclaurin_all_lt_objl 1);

by Auto_tac;

qed "Maclaurin_exp_lt";

Goal "EX t. abs t <= abs x & \

\           exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) + \

\                      (exp t / real (fact n)) * x ^ n";

by (cut_inst_tac [("diff","%n. exp"),("f","exp"),("x","x"),("n","n")] 

    Maclaurin_all_le_objl 1);

by Auto_tac;

qed "Maclaurin_exp_le";

(* ------------------------------------------------------------------------- *)

(* Version for sin function                                                  *)

(* ------------------------------------------------------------------------- *)

Goal "[| a < b; ALL x. a <= x & x <= b --> DERIV f x :> f'(x) |] \

\     ==> EX z. a < z & z < b & (f b - f a = (b - a) * f'(z))";

by (dtac MVT 1);

by (blast_tac (claset() addIs [DERIV_isCont]) 1);

by (force_tac (claset() addDs [order_less_imp_le],

    simpset() addsimps [differentiable_def]) 1);

by (blast_tac (claset() addDs [DERIV_unique,order_less_imp_le]) 1);

qed "MVT2";

Goal "d < (4::nat) ==> d = 0 | d = 1 | d = 2 | d = 3";

by (case_tac "d" 1 THEN Auto_tac);

qed "lemma_exhaust_less_4";

bind_thm ("real_mult_le_lemma",

          simplify (simpset()) (inst "b" "1" mult_right_mono));

Goal "0 < n --> Suc (Suc (2 * n - 2)) = 2*n";

by (induct_tac "n" 1);

by Auto_tac;

qed_spec_mp "Suc_Suc_mult_two_diff_two";

Addsimps [Suc_Suc_mult_two_diff_two];

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

by (induct_tac "n" 1);

by Auto_tac;

qed_spec_mp "lemma_Suc_Suc_4n_diff_2";

Addsimps [lemma_Suc_Suc_4n_diff_2];

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

by (induct_tac "n" 1);

by Auto_tac;

qed_spec_mp "Suc_mult_two_diff_one";

Addsimps [Suc_mult_two_diff_one];

Goal "EX t. sin x = \

\      (sumr 0 n (%m. (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)";

by (cut_inst_tac [("f","sin"),("n","n"),("x","x"),

       ("diff","%n x. sin(x + 1/2*real (n)*pi)")] 

       Maclaurin_all_lt_objl 1);

by (Safe_tac);

by (Simp_tac 1);

by (Simp_tac 1);

by (case_tac "n" 1);

by (Clarify_tac 1); 

by (Asm_full_simp_tac 1);

by (dres_inst_tac [("x","0")] spec 1 THEN Asm_full_simp_tac 1);

by (Asm_full_simp_tac 1);

by (rtac ccontr 1);

by (Asm_full_simp_tac 1);

by (dres_inst_tac [("x","x")] spec 1 THEN Asm_full_simp_tac 1);

by (dtac ssubst 1 THEN assume_tac 2);

by (res_inst_tac [("x","t")] exI 1);

by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);

by (rtac sumr_fun_eq 1);

by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));

by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex] delsimps [fact_Suc,realpow_Suc]));

(*Could sin_zero_iff help?*)

qed "Maclaurin_sin_expansion";

Goal "EX t. abs t <= abs x &  \

\      sin x = \

\      (sumr 0 n (%m. (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)";

by (cut_inst_tac [("f","sin"),("n","n"),("x","x"),

       ("diff","%n x. sin(x + 1/2*real (n)*pi)")] 

       Maclaurin_all_lt_objl 1);

by (Step_tac 1);

by (Simp_tac 1);

by (Simp_tac 1);

by (case_tac "n" 1);

by (Clarify_tac 1); 

by (Asm_full_simp_tac 1);

by (Asm_full_simp_tac 1);

by (rtac ccontr 1);

by (Asm_full_simp_tac 1);

by (dres_inst_tac [("x","x")] spec 1 THEN Asm_full_simp_tac 1);

by (dtac ssubst 1 THEN assume_tac 2);

by (res_inst_tac [("x","t")] exI 1);

by (rtac conjI 1);

by (arith_tac 1);

by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);

by (rtac sumr_fun_eq 1);

by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));

by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex] delsimps [fact_Suc,realpow_Suc]));

qed "Maclaurin_sin_expansion2";

Goal "[| 0 < n; 0 < x |] ==> \

\      EX t. 0 < t & t < x & \

\      sin x = \

\      (sumr 0 n (%m. (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)";

by (cut_inst_tac [("f","sin"),("n","n"),("h","x"),

       ("diff","%n x. sin(x + 1/2*real (n)*pi)")] 

       Maclaurin_objl 1);

by (Step_tac 1);

by (Asm_full_simp_tac 1);

by (Simp_tac 1);

by (dtac ssubst 1 THEN assume_tac 2);

by (res_inst_tac [("x","t")] exI 1);

by (rtac conjI 1 THEN rtac conjI 2);

by (assume_tac 1 THEN assume_tac 1);

by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);

by (rtac sumr_fun_eq 1);

by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));

by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex] delsimps [fact_Suc,realpow_Suc]));

qed "Maclaurin_sin_expansion3";

Goal "0 < x ==> \

\      EX t. 0 < t & t <= x & \

\      sin x = \

\      (sumr 0 n (%m. (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)";

by (cut_inst_tac [("f","sin"),("n","n"),("h","x"),

       ("diff","%n x. sin(x + 1/2*real (n)*pi)")] 

       Maclaurin2_objl 1);

by (Step_tac 1);

by (Asm_full_simp_tac 1);

by (Simp_tac 1);

by (dtac ssubst 1 THEN assume_tac 2);

by (res_inst_tac [("x","t")] exI 1);

by (rtac conjI 1 THEN rtac conjI 2);

by (assume_tac 1 THEN assume_tac 1);

by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);

by (rtac sumr_fun_eq 1);

by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));

by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex] delsimps [fact_Suc,realpow_Suc]));

qed "Maclaurin_sin_expansion4";

(*-----------------------------------------------------------------------------*)

(* Maclaurin expansion for cos                                                 *)

(*-----------------------------------------------------------------------------*)

Goal "sumr 0 (Suc n) \

\        (%m. (if even m \

\              then (- 1) ^ (m div 2)/(real  (fact m)) \

\              else 0) * \

\             0 ^ m) = 1";

by (induct_tac "n" 1);

by Auto_tac;

qed "sumr_cos_zero_one";

Addsimps [sumr_cos_zero_one];

Goal "EX t. abs t <= abs x & \

\      cos x = \

\      (sumr 0 n (%m. (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)";

by (cut_inst_tac [("f","cos"),("n","n"),("x","x"),

       ("diff","%n x. cos(x + 1/2*real (n)*pi)")] 

       Maclaurin_all_lt_objl 1);

by (Step_tac 1);

by (Simp_tac 1);

by (Simp_tac 1);

by (case_tac "n" 1);

by (Asm_full_simp_tac 1);

by (asm_full_simp_tac (simpset() delsimps [sumr_Suc]) 1);

by (rtac ccontr 1);

by (Asm_full_simp_tac 1);

by (dres_inst_tac [("x","x")] spec 1 THEN Asm_full_simp_tac 1);

by (dtac ssubst 1 THEN assume_tac 2);

by (res_inst_tac [("x","t")] exI 1);

by (rtac conjI 1);

by (arith_tac 1);

by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);

by (rtac sumr_fun_eq 1);

by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));

by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex,left_distrib,cos_add]  delsimps 

    [fact_Suc,realpow_Suc]));

by (auto_tac (claset(),simpset() addsimps [real_mult_commute]));

qed "Maclaurin_cos_expansion";

Goal "[| 0 < x; 0 < n |] ==> \

\      EX t. 0 < t & t < x & \

\      cos x = \

\      (sumr 0 n (%m. (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)";

by (cut_inst_tac [("f","cos"),("n","n"),("h","x"),

       ("diff","%n x. cos(x + 1/2*real (n)*pi)")] 

       Maclaurin_objl 1);

by (Step_tac 1);

by (Asm_full_simp_tac 1);

by (Simp_tac 1);

by (dtac ssubst 1 THEN assume_tac 2);

by (res_inst_tac [("x","t")] exI 1);

by (rtac conjI 1 THEN rtac conjI 2);

by (assume_tac 1 THEN assume_tac 1);

by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);

by (rtac sumr_fun_eq 1);

by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));

by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex,left_distrib,cos_add]  delsimps [fact_Suc,realpow_Suc]));

by (auto_tac (claset(),simpset() addsimps [real_mult_commute]));

qed "Maclaurin_cos_expansion2";

Goal "[| x < 0; 0 < n |] ==> \

\      EX t. x < t & t < 0 & \

\      cos x = \

\      (sumr 0 n (%m. (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)";

by (cut_inst_tac [("f","cos"),("n","n"),("h","x"),

       ("diff","%n x. cos(x + 1/2*real (n)*pi)")] 

       Maclaurin_minus_objl 1);

by (Step_tac 1);

by (Asm_full_simp_tac 1);

by (Simp_tac 1);

by (dtac ssubst 1 THEN assume_tac 2);

by (res_inst_tac [("x","t")] exI 1);

by (rtac conjI 1 THEN rtac conjI 2);

by (assume_tac 1 THEN assume_tac 1);

by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);

by (rtac sumr_fun_eq 1);

by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));

by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex,left_distrib,cos_add]  delsimps [fact_Suc,realpow_Suc]));

by (auto_tac (claset(),simpset() addsimps [real_mult_commute]));

qed "Maclaurin_minus_cos_expansion";

(* ------------------------------------------------------------------------- *)

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

(* ------------------------------------------------------------------------- *)