--- a/src/HOL/Hyperreal/MacLaurin.ML Tue May 11 14:00:02 2004 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,717 +0,0 @@
-(* 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 "abs(sin x - \
-\ sumr 0 n (%m. (if even m then 0 \
-\ else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) * \
-\ x ^ m)) \
-\ <= inverse(real (fact n)) * abs(x) ^ n";
-by (cut_inst_tac [("f","sin"),("n","n"),("x","x"),
- ("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)")] Maclaurin_all_le_objl 1);
-by (Step_tac 1);
-by (Asm_full_simp_tac 1);
-by (stac mod_Suc_eq_Suc_mod 1);
-by (cut_inst_tac [("m1","m")] (CLAIM "0 < (4::nat)" RS mod_less_divisor
- RS lemma_exhaust_less_4) 1);
-by (Step_tac 1);
-by (Asm_simp_tac 1);
-by (Asm_simp_tac 1);
-by (Asm_simp_tac 1);
-by (rtac DERIV_minus 1 THEN Simp_tac 1);
-by (Asm_simp_tac 1);
-by (rtac lemma_DERIV_subst 1 THEN rtac DERIV_minus 1 THEN rtac DERIV_cos 1);
-by (Simp_tac 1);
-by (dtac ssubst 1 THEN assume_tac 2);
-by (rtac (ARITH_PROVE "[|x = y; abs u <= (v::real) |] ==> abs ((x + u) - y) <= v") 1);
-by (rtac sumr_fun_eq 1);
-by (Step_tac 1);
-by (rtac (CLAIM "x = y ==> x * z = y * (z::real)") 1);
-by (stac even_even_mod_4_iff 1);
-by (cut_inst_tac [("m1","r")] (CLAIM "0 < (4::nat)" RS mod_less_divisor
- RS lemma_exhaust_less_4) 1);
-by (Step_tac 1);
-by (Asm_simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [even_num_iff]) 2);
-by (asm_simp_tac (simpset() addsimps [even_num_iff]) 1);
-by (asm_simp_tac (simpset() addsimps [even_num_iff]) 2);
-by (dtac lemma_even_mod_4_div_2 1);
-by (asm_full_simp_tac (simpset() addsimps [numeral_2_eq_2,real_divide_def]) 1);
-by (dtac lemma_odd_mod_4_div_2 1);
-by (asm_full_simp_tac (simpset() addsimps [numeral_2_eq_2, real_divide_def]) 1);
-by (auto_tac (claset() addSIs [real_mult_le_lemma,mult_right_mono],
- simpset() addsimps [real_divide_def,abs_mult,abs_inverse,power_abs RS
-sym]));
-qed "Maclaurin_sin_bound";
-
-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?? *)
-(* ------------------------------------------------------------------------- *)
-