--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Hyperreal/Transcendental.ML Thu Nov 15 16:12:49 2001 +0100
@@ -0,0 +1,2515 @@
+(* Title : Transcendental.ML
+ Author : Jacques D. Fleuriot
+ Copyright : 1998,1999 University of Cambridge
+ 1999 University of Edinburgh
+ Description : Power Series
+*)
+
+Goalw [root_def] "root (Suc n) 0 = 0";
+by (safe_tac (claset() addSIs [some_equality,realpow_zero]
+ addSEs [realpow_zero_zero]));
+qed "real_root_zero";
+Addsimps [real_root_zero];
+
+Goalw [root_def]
+ "0 < x ==> (root(Suc n) x) ^ (Suc n) = x";
+by (dres_inst_tac [("n","n")] realpow_pos_nth2 1);
+by (auto_tac (claset() addIs [someI2],simpset()));
+qed "real_root_pow_pos";
+
+Goal "0 <= x ==> (root(Suc n) x) ^ (Suc n) = x";
+by (auto_tac (claset() addSDs [real_le_imp_less_or_eq]
+ addDs [real_root_pow_pos],simpset()));
+qed "real_root_pow_pos2";
+
+Goalw [root_def]
+ "0 < x ==> root(Suc n) (x ^ (Suc n)) = x";
+by (rtac some_equality 1);
+by (forw_inst_tac [("n","n")] realpow_gt_zero 2);
+by (auto_tac (claset(),simpset() addsimps [real_0_less_mult_iff]));
+by (res_inst_tac [("R1.0","u"),("R2.0","x")] real_linear_less2 1);
+by (dres_inst_tac [("n3","n"),("x","u")]
+ (zero_less_Suc RSN (3,conjI RSN (2,conjI RS realpow_less))) 1);
+by (dres_inst_tac [("n3","n"),("x","x")]
+ (zero_less_Suc RSN (3,conjI RSN (2,conjI RS realpow_less))) 4);
+by (auto_tac (claset(),simpset() addsimps [real_less_not_refl]));
+qed "real_root_pos";
+
+Goal "0 <= x ==> root(Suc n) (x ^ (Suc n)) = x";
+by (auto_tac (claset() addSDs [real_le_imp_less_or_eq,
+ real_root_pos],simpset()));
+qed "real_root_pos2";
+
+Goalw [root_def]
+ "0 < x ==> 0 <= root(Suc n) x";
+by (dres_inst_tac [("n","n")] realpow_pos_nth2 1);
+by (Safe_tac THEN rtac someI2 1);
+by (auto_tac (claset() addSIs [order_less_imp_le]
+ addDs [realpow_gt_zero],simpset() addsimps [real_0_less_mult_iff]));
+qed "real_root_pos_pos";
+
+Goal "0 <= x ==> 0 <= root(Suc n) x";
+by (auto_tac (claset() addSDs [real_le_imp_less_or_eq]
+ addDs [real_root_pos_pos],simpset()));
+qed "real_root_pos_pos_le";
+
+Goalw [root_def] "root (Suc n) 1 = 1";
+by (rtac some_equality 1);
+by Auto_tac;
+by (rtac ccontr 1);
+by (res_inst_tac [("R1.0","u"),("R2.0","1")] real_linear_less2 1);
+by (dres_inst_tac [("n","n")] realpow_Suc_less_one 1);
+by (dres_inst_tac [("n","n")] realpow_Suc_gt_one 4);
+by (auto_tac (claset(),simpset() addsimps [real_less_not_refl]));
+qed "real_root_one";
+Addsimps [real_root_one];
+
+(*----------------------------------------------------------------------*)
+(* Square root *)
+(*----------------------------------------------------------------------*)
+
+(*lcp: needed now because 2 is a binary numeral!*)
+Goal "root 2 = root (Suc (Suc 0))";
+by (simp_tac (simpset() delsimps [numeral_0_eq_0, numeral_1_eq_1]
+ addsimps [numeral_0_eq_0 RS sym]) 1);
+qed "root_2_eq";
+Addsimps [root_2_eq];
+
+Goalw [sqrt_def] "sqrt 0 = 0";
+by (Auto_tac);
+qed "real_sqrt_zero";
+Addsimps [real_sqrt_zero];
+
+Goalw [sqrt_def] "sqrt 1 = 1";
+by (Auto_tac);
+qed "real_sqrt_one";
+Addsimps [real_sqrt_one];
+
+Goalw [sqrt_def] "(sqrt(x) ^ 2 = x) = (0 <= x)";
+by (Step_tac 1);
+by (cut_inst_tac [("r","root 2 x")] realpow_two_le 1);
+by (stac numeral_2_eq_2 2);
+by (rtac real_root_pow_pos2 2);
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "real_sqrt_pow2_iff";
+Addsimps [real_sqrt_pow2_iff];
+
+
+Addsimps [realpow_two_le_add_order RS (real_sqrt_pow2_iff RS iffD2)];
+Addsimps [simplify (simpset()) (realpow_two_le_add_order RS
+ (real_sqrt_pow2_iff RS iffD2))];
+
+Goalw [sqrt_def] "0 < x ==> sqrt(x) ^ 2 = x";
+by (stac numeral_2_eq_2 1);
+by (etac real_root_pow_pos 1);
+qed "real_sqrt_gt_zero_pow2";
+
+Goal "(sqrt(abs(x)) ^ 2 = abs x)";
+by (rtac (real_sqrt_pow2_iff RS iffD2) 1);
+by (arith_tac 1);
+qed "real_sqrt_abs_abs";
+Addsimps [real_sqrt_abs_abs];
+
+Goalw [sqrt_def]
+ "0 <= x ==> sqrt(x) ^ 2 = sqrt(x ^ 2)";
+by (stac numeral_2_eq_2 1);
+by (auto_tac (claset() addIs [real_root_pow_pos2
+ RS ssubst, real_root_pos2 RS ssubst],
+ simpset() delsimps [realpow_Suc]));
+qed "real_pow_sqrt_eq_sqrt_pow";
+
+Goal "0 <= x ==> sqrt(x) ^ 2 = sqrt(abs(x ^ 2))";
+by (asm_full_simp_tac (simpset() addsimps [real_pow_sqrt_eq_sqrt_pow]) 1);
+by (stac numeral_2_eq_2 1);
+by (asm_full_simp_tac (simpset()delsimps [realpow_Suc]) 1);
+qed "real_pow_sqrt_eq_sqrt_abs_pow";
+
+Goal "0 <= x ==> sqrt(x) ^ 2 = sqrt(abs(x) ^ 2)";
+by (asm_full_simp_tac (simpset() addsimps [real_pow_sqrt_eq_sqrt_abs_pow]) 1);
+by (stac numeral_2_eq_2 1);
+by (asm_full_simp_tac (simpset()delsimps [realpow_Suc]) 1);
+qed "real_pow_sqrt_eq_sqrt_abs_pow2";
+
+Goal "0 <= x ==> sqrt(x) ^ 2 = abs(x)";
+by (rtac (real_sqrt_abs_abs RS subst) 1);
+by (res_inst_tac [("x1","x")]
+ (real_pow_sqrt_eq_sqrt_abs_pow2 RS ssubst) 1);
+by (rtac (real_pow_sqrt_eq_sqrt_pow RS sym) 2);
+by (assume_tac 1 THEN arith_tac 1);
+qed "real_sqrt_pow_abs";
+
+Goal "(~ (0::real) < x*x) = (x = 0)";
+by Auto_tac;
+by (rtac ccontr 1);
+by (cut_inst_tac [("R1.0","x"),("R2.0","0")] real_linear 1);
+by Auto_tac;
+by (ftac (real_mult_order) 2);
+by (asm_full_simp_tac (simpset() addsimps [real_0_less_mult_iff]) 1);
+by Auto_tac;
+qed "not_real_square_gt_zero";
+Addsimps [not_real_square_gt_zero];
+
+
+(* proof used to be simpler *)
+Goalw [sqrt_def,root_def]
+ "[| 0 < x; 0 < y |] ==>sqrt(x*y) = sqrt(x) * sqrt(y)";
+by (dres_inst_tac [("n","1")] realpow_pos_nth2 1);
+by (dres_inst_tac [("n","1")] realpow_pos_nth2 1);
+by (asm_full_simp_tac (simpset() delsimps [realpow_Suc]
+ addsimps [numeral_2_eq_2]) 1);
+by (Step_tac 1);
+by (rtac someI2 1 THEN Step_tac 1 THEN Blast_tac 2);
+by (Asm_full_simp_tac 1 THEN Asm_full_simp_tac 1);
+by (rtac someI2 1 THEN Step_tac 1 THEN Blast_tac 2);
+by (Asm_full_simp_tac 1 THEN Asm_full_simp_tac 1);
+by (res_inst_tac [("a","xa * x")] someI2 1);
+by (auto_tac (claset() addEs [real_less_asym],
+ simpset() addsimps real_mult_ac@[realpow_mult RS sym,realpow_two_disj,
+ realpow_gt_zero, real_mult_order] delsimps [realpow_Suc]));
+qed "real_sqrt_mult_distrib";
+
+Goal "[|0<=x; 0<=y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)";
+by (auto_tac (claset() addIs [ real_sqrt_mult_distrib],
+ simpset() addsimps [real_le_less]));
+qed "real_sqrt_mult_distrib2";
+
+Goal "(r * r = 0) = (r = (0::real))";
+by Auto_tac;
+qed "real_mult_self_eq_zero_iff";
+Addsimps [real_mult_self_eq_zero_iff];
+
+Goalw [sqrt_def,root_def] "0 < x ==> 0 < sqrt(x)";
+by (stac numeral_2_eq_2 1);
+by (dtac realpow_pos_nth2 1 THEN Step_tac 1);
+by (rtac someI2 1 THEN Step_tac 1 THEN Blast_tac 2);
+by Auto_tac;
+qed "real_sqrt_gt_zero";
+
+Goal "0 <= x ==> 0 <= sqrt(x)";
+by (auto_tac (claset() addIs [real_sqrt_gt_zero],
+ simpset() addsimps [real_le_less]));
+qed "real_sqrt_ge_zero";
+
+Goal "0 <= sqrt (x ^ 2 + y ^ 2)";
+by (auto_tac (claset() addSIs [real_sqrt_ge_zero],simpset()));
+qed "real_sqrt_sum_squares_ge_zero";
+Addsimps [real_sqrt_sum_squares_ge_zero];
+
+Goal "0 <= sqrt ((x ^ 2 + y ^ 2)*(xa ^ 2 + ya ^ 2))";
+by (auto_tac (claset() addSIs [real_sqrt_ge_zero],simpset()
+ addsimps [real_0_le_mult_iff]));
+qed "real_sqrt_sum_squares_mult_ge_zero";
+Addsimps [real_sqrt_sum_squares_mult_ge_zero];
+
+Goal "sqrt ((x ^ 2 + y ^ 2) * (xa ^ 2 + ya ^ 2)) ^ 2 = \
+\ (x ^ 2 + y ^ 2) * (xa ^ 2 + ya ^ 2)";
+by (auto_tac (claset(),simpset() addsimps [real_sqrt_pow2_iff,
+ real_0_le_mult_iff] delsimps [realpow_Suc]));
+qed "real_sqrt_sum_squares_mult_squared_eq";
+Addsimps [real_sqrt_sum_squares_mult_squared_eq];
+
+Goal "sqrt(x ^ 2) = abs(x)";
+by (rtac (abs_realpow_two RS subst) 1);
+by (rtac (real_sqrt_abs_abs RS subst) 1);
+by (rtac (real_pow_sqrt_eq_sqrt_pow RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2, abs_mult,abs_ge_zero]));
+qed "real_sqrt_abs";
+Addsimps [real_sqrt_abs];
+
+Goal "sqrt(x*x) = abs(x)";
+by (rtac (realpow_two RS subst) 1);
+by (stac (numeral_2_eq_2 RS sym) 1);
+by (rtac real_sqrt_abs 1);
+qed "real_sqrt_abs2";
+Addsimps [real_sqrt_abs2];
+
+Goal "0 < x ==> 0 < sqrt(x) ^ 2";
+by (asm_full_simp_tac (simpset() addsimps [real_sqrt_gt_zero_pow2]) 1);
+qed "real_sqrt_pow2_gt_zero";
+
+Goal "0 < x ==> sqrt x ~= 0";
+by (forward_tac [real_sqrt_pow2_gt_zero] 1);
+by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2, real_less_not_refl]));
+qed "real_sqrt_not_eq_zero";
+
+Goal "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x";
+by (forward_tac [real_sqrt_not_eq_zero] 1);
+by (cut_inst_tac [("n1","2"),("r1","sqrt x")] (realpow_inverse RS sym) 1);
+by (auto_tac (claset() addDs [real_sqrt_gt_zero_pow2],simpset()));
+qed "real_inv_sqrt_pow2";
+
+Goal "[| 0 <= x; sqrt(x) = 0|] ==> x = 0";
+by (dtac real_le_imp_less_or_eq 1);
+by (auto_tac (claset() addDs [real_sqrt_not_eq_zero],simpset()));
+qed "real_sqrt_eq_zero_cancel";
+
+Goal "0 <= x ==> ((sqrt x = 0) = (x = 0))";
+by (auto_tac (claset(),simpset() addsimps [real_sqrt_eq_zero_cancel]));
+qed "real_sqrt_eq_zero_cancel_iff";
+Addsimps [real_sqrt_eq_zero_cancel_iff];
+
+Goal "x <= sqrt(x ^ 2 + y ^ 2)";
+by (subgoal_tac "x <= 0 | 0 <= x" 1);
+by (Step_tac 1);
+by (rtac real_le_trans 1);
+by (auto_tac (claset(),simpset() delsimps [realpow_Suc]));
+by (res_inst_tac [("n","1")] realpow_increasing 1);
+by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2 RS sym]
+ delsimps [realpow_Suc]));
+qed "real_sqrt_sum_squares_ge1";
+Addsimps [real_sqrt_sum_squares_ge1];
+
+Goal "y <= sqrt(z ^ 2 + y ^ 2)";
+by (simp_tac (simpset() addsimps [real_add_commute]
+ delsimps [realpow_Suc]) 1);
+qed "real_sqrt_sum_squares_ge2";
+Addsimps [real_sqrt_sum_squares_ge2];
+
+Goal "1 <= x ==> 1 <= sqrt x";
+by (res_inst_tac [("n","1")] realpow_increasing 1);
+by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2 RS sym, real_sqrt_gt_zero_pow2,
+ real_sqrt_ge_zero] delsimps [realpow_Suc]));
+qed "real_sqrt_ge_one";
+
+(*-------------------------------------------------------------------------*)
+(* Exponential function *)
+(*-------------------------------------------------------------------------*)
+
+Goal "summable (%n. inverse (real (fact n)) * x ^ n)";
+by (cut_facts_tac [real_zero_less_one RS real_dense] 1);
+by (Step_tac 1);
+by (cut_inst_tac [("x","r")] reals_Archimedean3 1);
+by Auto_tac;
+by (dres_inst_tac [("x","abs x")] spec 1 THEN Step_tac 1);
+by (res_inst_tac [("N","n"),("c","r")] ratio_test 1);
+by (auto_tac (claset(),simpset() addsimps [abs_mult,real_mult_assoc RS sym]
+ delsimps [fact_Suc]));
+by (rtac real_mult_le_le_mono2 1);
+by (res_inst_tac [("w1","abs x")] (real_mult_commute RS ssubst) 2);
+by (rtac (fact_Suc RS ssubst) 2);
+by (rtac (real_of_nat_mult RS ssubst) 2);
+by (auto_tac (claset(),simpset() addsimps [abs_mult,real_inverse_distrib,
+ abs_ge_zero]));
+by (auto_tac (claset(), simpset() addsimps
+ [real_mult_assoc RS sym, abs_ge_zero, abs_eqI2,
+ real_inverse_gt_0]));
+by (rtac (CLAIM "x < (y::real) ==> x <= y") 1);
+by (res_inst_tac [("z1","real (Suc na)")] (real_mult_less_iff1
+ RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [real_not_refl2 RS not_sym,
+ real_mult_assoc,abs_inverse]));
+by (rtac real_less_trans 1);
+by (auto_tac (claset(),simpset() addsimps real_mult_ac));
+qed "summable_exp";
+
+Addsimps [real_of_nat_fact_gt_zero,
+ real_of_nat_fact_ge_zero,inv_real_of_nat_fact_gt_zero,
+ inv_real_of_nat_fact_ge_zero];
+
+Goalw [real_divide_def]
+ "summable (%n. \
+\ (if even n then 0 \
+\ else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * \
+\ x ^ n)";
+by (res_inst_tac [("g","(%n. inverse (real (fact n)) * abs(x) ^ n)")]
+ summable_comparison_test 1);
+by (rtac summable_exp 2);
+by (res_inst_tac [("x","0")] exI 1);
+by (auto_tac (claset(), simpset() addsimps [realpow_abs,
+ abs_ge_zero,abs_mult,real_0_le_mult_iff]));
+by (auto_tac (claset() addIs [real_mult_le_le_mono2],
+ simpset() addsimps [real_inverse_gt_0,abs_eqI2,abs_ge_zero]));
+qed "summable_sin";
+
+Goalw [real_divide_def]
+ "summable (%n. \
+\ (if even n then \
+\ (- 1) ^ (n div 2)/(real (fact n)) else 0) * x ^ n)";
+by (res_inst_tac [("g","(%n. inverse (real (fact n)) * abs(x) ^ n)")]
+ summable_comparison_test 1);
+by (rtac summable_exp 2);
+by (res_inst_tac [("x","0")] exI 1);
+by (auto_tac (claset(), simpset() addsimps [realpow_abs,abs_ge_zero,abs_mult,
+ real_0_le_mult_iff]));
+by (auto_tac (claset() addSIs [real_mult_le_le_mono2],
+ simpset() addsimps [real_inverse_gt_0,abs_eqI2,abs_ge_zero]));
+qed "summable_cos";
+
+Goal "(if even n then 0 \
+\ else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0";
+by (induct_tac "n" 1);
+by (Auto_tac);
+val lemma_STAR_sin = result();
+Addsimps [lemma_STAR_sin];
+
+Goal "0 < n --> \
+\ (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0";
+by (induct_tac "n" 1);
+by (Auto_tac);
+val lemma_STAR_cos = result();
+Addsimps [lemma_STAR_cos];
+
+Goal "0 < n --> \
+\ (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0";
+by (induct_tac "n" 1);
+by (Auto_tac);
+val lemma_STAR_cos1 = result();
+Addsimps [lemma_STAR_cos1];
+
+Goal "sumr 1 n (%n. if even n \
+\ then (- 1) ^ (n div 2)/(real (fact n)) * \
+\ 0 ^ n \
+\ else 0) = 0";
+by (induct_tac "n" 1);
+by (case_tac "n" 2);
+by (Auto_tac);
+val lemma_STAR_cos2 = result();
+Addsimps [lemma_STAR_cos2];
+
+Goalw [exp_def] "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)";
+by (rtac (summable_exp RS summable_sums) 1);
+qed "exp_converges";
+
+Goalw [sin_def]
+ "(%n. (if even n then 0 \
+\ else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * \
+\ x ^ n) sums sin(x)";
+by (rtac (summable_sin RS summable_sums) 1);
+qed "sin_converges";
+
+Goalw [cos_def]
+ "(%n. (if even n then \
+\ (- 1) ^ (n div 2)/(real (fact n)) \
+\ else 0) * x ^ n) sums cos(x)";
+by (rtac (summable_cos RS summable_sums) 1);
+qed "cos_converges";
+
+Goal "p <= n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y";
+by (induct_tac "n" 1 THEN Auto_tac);
+by (subgoal_tac "p = Suc n" 1);
+by (Asm_simp_tac 1 THEN Auto_tac);
+by (dtac sym 1 THEN asm_full_simp_tac (simpset() addsimps
+ [Suc_diff_le,real_mult_commute,realpow_Suc RS sym]
+ delsimps [realpow_Suc]) 1);
+qed_spec_mp "lemma_realpow_diff";
+
+(*--------------------------------------------------------------------------*)
+(* Properties of power series *)
+(*--------------------------------------------------------------------------*)
+
+Goal "sumr 0 (Suc n) (%p. (x ^ p) * y ^ ((Suc n) - p)) = \
+\ y * sumr 0 (Suc n) (%p. (x ^ p) * (y ^ (n - p)))";
+by (auto_tac (claset(),simpset() addsimps [sumr_mult] delsimps [sumr_Suc]));
+by (rtac sumr_subst 1);
+by (strip_tac 1);
+by (rtac (lemma_realpow_diff RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps real_mult_ac));
+qed "lemma_realpow_diff_sumr";
+
+Goal "x ^ (Suc n) - y ^ (Suc n) = \
+\ (x - y) * sumr 0 (Suc n) (%p. (x ^ p) * (y ^(n - p)))";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() delsimps [sumr_Suc]));
+by (rtac (sumr_Suc RS ssubst) 1);
+by (dtac sym 1);
+by (auto_tac (claset(),simpset() addsimps [lemma_realpow_diff_sumr,
+ real_add_mult_distrib2,real_diff_def] @
+ real_mult_ac delsimps [sumr_Suc]));
+qed "lemma_realpow_diff_sumr2";
+
+Goal "sumr 0 (Suc n) (%p. (x ^ p) * (y ^ (n - p))) = \
+\ sumr 0 (Suc n) (%p. (x ^ (n - p)) * (y ^ p))";
+by (case_tac "x = y" 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_commute,
+ realpow_add RS sym] delsimps [sumr_Suc]));
+by (res_inst_tac [("c1","x - y")] (real_mult_left_cancel RS iffD1) 1);
+by (rtac (real_minus_minus RS subst) 2);
+by (rtac (real_minus_mult_eq1 RS ssubst) 2);
+by (auto_tac (claset(),simpset() addsimps [lemma_realpow_diff_sumr2
+ RS sym] delsimps [sumr_Suc]));
+qed "lemma_realpow_rev_sumr";
+
+(* ------------------------------------------------------------------------ *)
+(* Power series has a `circle` of convergence, *)
+(* i.e. if it sums for x, then it sums absolutely for z with |z| < |x|. *)
+(* ------------------------------------------------------------------------ *)
+
+Goalw [real_divide_def] "1/(x::real) = inverse x";
+by (Simp_tac 1);
+qed "real_divide_eq_inverse";
+
+Goal "[| summable (%n. f(n) * (x ^ n)); abs(z) < abs(x) |] \
+\ ==> summable (%n. abs(f(n)) * (z ^ n))";
+by (dtac summable_LIMSEQ_zero 1);
+by (dtac convergentI 1);
+by (asm_full_simp_tac (simpset() addsimps [Cauchy_convergent_iff RS sym]) 1);
+by (dtac Cauchy_Bseq 1);
+by (asm_full_simp_tac (simpset() addsimps [Bseq_def]) 1);
+by (Step_tac 1);
+by (res_inst_tac [("g","%n. K * abs(z ^ n) * inverse (abs(x ^ n))")]
+ summable_comparison_test 1);
+by (res_inst_tac [("x","0")] exI 1 THEN Step_tac 1);
+by (subgoal_tac "0 < abs (x ^ n)" 1);
+by (res_inst_tac [("z","abs (x ^ n)")] (CLAIM_SIMP
+ "[| (0::real) <z; x*z<=y*z |] ==> x<=y" [real_mult_le_cancel1]) 1);
+by (auto_tac (claset(),
+ simpset() addsimps [real_mult_assoc,realpow_abs RS sym]));
+by (dres_inst_tac [("x","0")] spec 2 THEN Force_tac 2);
+by (auto_tac (claset(),simpset() addsimps [abs_ge_zero,
+ abs_mult,realpow_abs RS sym] @ real_mult_ac));
+by (res_inst_tac [("x2","z ^ n")] (abs_ge_zero RS real_le_imp_less_or_eq
+ RS disjE) 1 THEN dtac sym 2);
+by (auto_tac (claset() addSIs [real_mult_le_le_mono2],
+ simpset() addsimps [real_mult_assoc RS sym,
+ realpow_abs RS sym,summable_def]));
+by (res_inst_tac [("x","K * inverse(1 - (abs(z) * inverse(abs x)))")] exI 1);
+by (auto_tac (claset() addSIs [sums_mult],simpset() addsimps [real_mult_assoc]));
+by (subgoal_tac
+ "abs(z ^ n) * inverse(abs x ^ n) = (abs(z) * inverse(abs x)) ^ n" 1);
+by (auto_tac (claset(),simpset() addsimps [realpow_abs]));
+by (subgoal_tac "x ~= 0" 1);
+by (subgoal_tac "x ~= 0" 3);
+by (auto_tac (claset(),simpset() addsimps
+ [abs_inverse RS sym,realpow_not_zero,abs_mult
+ RS sym,realpow_inverse,realpow_mult RS sym]));
+by (auto_tac (claset() addSIs [geometric_sums],simpset() addsimps
+ [realpow_abs RS sym,real_divide_eq_inverse RS sym]));
+by (res_inst_tac [("z","abs(x)")] (CLAIM_SIMP
+ "[|(0::real)<z; x*z<y*z |] ==> x<y" [real_mult_less_cancel1]) 1);
+by (auto_tac (claset(),simpset() addsimps [abs_ge_zero,
+ abs_mult RS sym,real_mult_assoc]));
+qed "powser_insidea";
+
+Goal "[| summable (%n. f(n) * (x ^ n)); abs(z) < abs(x) |] \
+\ ==> summable (%n. f(n) * (z ^ n))";
+by (dres_inst_tac [("z","abs z")] powser_insidea 1);
+by (auto_tac (claset() addIs [summable_rabs_cancel],
+ simpset() addsimps [realpow_abs,abs_mult RS sym]));
+qed "powser_inside";
+
+(* ------------------------------------------------------------------------ *)
+(* Differentiation of power series *)
+(* ------------------------------------------------------------------------ *)
+
+(* Lemma about distributing negation over it *)
+Goalw [diffs_def] "diffs (%n. - c n) = (%n. - diffs c n)";
+by Auto_tac;
+qed "diffs_minus";
+
+(* ------------------------------------------------------------------------ *)
+(* Show that we can shift the terms down one *)
+(* ------------------------------------------------------------------------ *)
+
+Goal "sumr 0 n (%n. (diffs c)(n) * (x ^ n)) = \
+\ sumr 0 n (%n. real n * c(n) * (x ^ (n - Suc 0))) + \
+\ (real n * c(n) * x ^ (n - Suc 0))";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc,
+ real_add_assoc RS sym,diffs_def]));
+qed "lemma_diffs";
+
+Goal "sumr 0 n (%n. real n * c(n) * (x ^ (n - Suc 0))) = \
+\ sumr 0 n (%n. (diffs c)(n) * (x ^ n)) - \
+\ (real n * c(n) * x ^ (n - Suc 0))";
+by (auto_tac (claset(),simpset() addsimps [lemma_diffs]));
+qed "lemma_diffs2";
+
+Goal "summable (%n. (diffs c)(n) * (x ^ n)) ==> \
+\ (%n. real n * c(n) * (x ^ (n - Suc 0))) sums \
+\ (suminf(%n. (diffs c)(n) * (x ^ n)))";
+by (ftac summable_LIMSEQ_zero 1);
+by (subgoal_tac "(%n. real n * c(n) * (x ^ (n - Suc 0))) ----> 0" 1);
+by (rtac LIMSEQ_imp_Suc 2);
+by (dtac summable_sums 1);
+by (auto_tac (claset(),simpset() addsimps [sums_def]));
+by (thin_tac "(%n. diffs c n * x ^ n) ----> 0" 1);
+by (rotate_tac 1 1);
+by (dtac LIMSEQ_diff 1);
+by (auto_tac (claset(),simpset() addsimps [lemma_diffs2 RS sym,
+ symmetric diffs_def]));
+by (asm_full_simp_tac (simpset() addsimps [diffs_def]) 1);
+qed "diffs_equiv";
+
+(* -------------------------------------------------------------------------*)
+(* Term-by-term differentiability of power series *)
+(* -------------------------------------------------------------------------*)
+
+Goal "sumr 0 m (%p. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) = \
+\ sumr 0 m (%p. (z ^ p) * \
+\ (((z + h) ^ (m - p)) - (z ^ (m - p))))";
+by (rtac sumr_subst 1);
+by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2,
+ real_diff_def,realpow_add RS sym]
+ @ real_mult_ac));
+qed "lemma_termdiff1";
+
+(* proved elsewhere? *)
+Goal "m < n --> (EX d. n = m + d + Suc 0)";
+by (induct_tac "m" 1 THEN Auto_tac);
+by (case_tac "n" 1);
+by (case_tac "d" 3);
+by (Auto_tac);
+qed_spec_mp "less_add_one";
+
+Goal " h ~= 0 ==> \
+\ (((z + h) ^ n) - (z ^ n)) * inverse h - \
+\ real n * (z ^ (n - Suc 0)) = \
+\ h * sumr 0 (n - Suc 0) (%p. (z ^ p) * \
+\ sumr 0 ((n - Suc 0) - p) \
+\ (%q. ((z + h) ^ q) * (z ^ (((n - 2) - p) - q))))";
+by (rtac (real_mult_left_cancel RS iffD1) 1 THEN Asm_simp_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [real_diff_mult_distrib2]
+ @ real_mult_ac) 1);
+by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc RS sym]) 1);
+by (case_tac "n" 1 THEN auto_tac (claset(),simpset()
+ addsimps [lemma_realpow_diff_sumr2,
+ real_diff_mult_distrib2 RS sym,real_mult_assoc]
+ delsimps [realpow_Suc,sumr_Suc]));
+by (rtac (real_mult_left_cancel RS iffD2) 1);
+by (auto_tac (claset(),simpset() addsimps [lemma_realpow_rev_sumr]
+ delsimps [sumr_Suc]));
+by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc,sumr_diff_mult_const,
+ real_add_mult_distrib,CLAIM "(a + b) - (c + d) = a - c + b - (d::real)",
+ lemma_termdiff1,sumr_mult]));
+by (auto_tac (claset() addSIs [sumr_subst],simpset() addsimps
+ [real_diff_def,real_add_assoc]));
+by (fold_tac [real_diff_def] THEN dtac less_add_one 1);
+by (auto_tac (claset(),simpset() addsimps [sumr_mult,lemma_realpow_diff_sumr2]
+ @ real_mult_ac delsimps [sumr_Suc,realpow_Suc]));
+qed "lemma_termdiff2";
+
+Goal "[| h ~= 0; abs z <= K; abs (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) * abs h";
+by (rtac (lemma_termdiff2 RS ssubst) 1);
+by (asm_full_simp_tac (simpset() addsimps [abs_mult,real_mult_commute]) 2);
+by (stac real_mult_commute 2);
+by (rtac (sumr_rabs RS real_le_trans) 2);
+by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc RS sym]) 2);
+by (rtac (real_mult_commute RS subst) 2);
+by (auto_tac (claset() addSIs [sumr_bound2],simpset() addsimps
+ [abs_ge_zero,abs_mult]));
+by (case_tac "n" 1 THEN Auto_tac);
+by (dtac less_add_one 1);
+by (auto_tac (claset(),simpset() addsimps [realpow_add,real_add_assoc RS sym,
+ CLAIM_SIMP "(a * b) * c = a * (c * (b::real))" real_mult_ac]
+ delsimps [sumr_Suc]));
+by (auto_tac (claset() addSIs [real_mult_le_mono],simpset() addsimps
+ [abs_ge_zero] delsimps [sumr_Suc]));
+by (auto_tac (claset() addSIs [realpow_le2],simpset() addsimps
+ [realpow_abs RS sym,abs_ge_zero] delsimps [sumr_Suc] ));
+by (res_inst_tac [("j","real (Suc d) * (K ^ d)")] real_le_trans 1);
+by (subgoal_tac "0 <= K" 2);
+by (arith_tac 3);
+by (dres_inst_tac [("n","d")] realpow_ge_zero2 2);
+by (auto_tac (claset(),simpset() addsimps
+ [real_of_nat_le_iff RS sym] delsimps [sumr_Suc] ));
+by (rtac (sumr_rabs RS real_le_trans) 1);
+by (rtac sumr_bound2 1 THEN auto_tac (claset() addSDs [less_add_one]
+ addSIs [real_mult_le_mono],simpset() addsimps [abs_mult,
+ realpow_add,abs_ge_zero]));
+by (auto_tac (claset() addSIs [realpow_le2,realpow_ge_zero],simpset() addsimps
+ [realpow_abs RS sym,abs_ge_zero]));
+by (ALLGOALS(arith_tac));
+qed "lemma_termdiff3";
+
+Addsimps [abs_ge_zero];
+
+Goalw [LIM_def]
+ "[| 0 < k; \
+\ (ALL h. 0 < abs(h) & abs(h) < k --> abs(f h) <= K * abs(h)) |] \
+\ ==> f -- 0 --> 0";
+by (Auto_tac);
+by (subgoal_tac "0 <= K" 1);
+by (dres_inst_tac [("x","k*inverse 2")] spec 2);
+by (ftac real_less_half_sum 2);
+by (dtac real_gt_half_sum 2);
+by (auto_tac (claset(),simpset() addsimps [abs_eqI2]));
+by (res_inst_tac [("z","k/2")] (CLAIM_SIMP
+ "[| (0::real) <z; x*z<=y*z |] ==> x<=y" [real_mult_le_cancel1]) 2);
+by (auto_tac (claset() addIs [abs_ge_zero RS real_le_trans],simpset()));
+by (dtac real_le_imp_less_or_eq 1);
+by Auto_tac;
+by (subgoal_tac "0 < (r * inverse K) * inverse 2" 1);
+by (REPEAT(rtac (real_mult_order) 2));
+by (dres_inst_tac [("d1.0","r * inverse K * inverse 2"),("d2.0","k")]
+ real_lbound_gt_zero 1);
+by (auto_tac (claset(),simpset() addsimps [real_inverse_gt_0,
+ real_0_less_mult_iff]));
+by (rtac real_le_trans 2 THEN assume_tac 3 THEN Auto_tac);
+by (res_inst_tac [("x","e")] exI 1 THEN Auto_tac);
+by (res_inst_tac [("y","K * abs x")] order_le_less_trans 1);
+by (res_inst_tac [("R2.0","K * e")] real_less_trans 2);
+by (res_inst_tac [("z","inverse K")] (CLAIM_SIMP
+ "[|(0::real) <z; z*x<z*y |] ==> x<y" [real_mult_less_cancel1]) 3);
+by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc RS sym]) 4);
+by Auto_tac;
+by (auto_tac (claset(),simpset() addsimps real_mult_ac));
+qed "lemma_termdiff4";
+
+Goal "[| 0 < k; \
+\ summable f; \
+\ ALL h. 0 < abs(h) & abs(h) < k --> \
+\ (ALL n. abs(g(h) (n::nat)) <= (f(n) * abs(h))) |] \
+\ ==> (%h. suminf(g h)) -- 0 --> 0";
+by (dtac summable_sums 1);
+by (subgoal_tac "ALL h. 0 < abs h & abs h < k --> \
+\ abs(suminf (g h)) <= suminf f * abs h" 1);
+by (Auto_tac);
+by (subgoal_tac "summable (%n. f n * abs h)" 2);
+by (simp_tac (simpset() addsimps [summable_def]) 3);
+by (res_inst_tac [("x","suminf f * abs h")] exI 3);
+by (dres_inst_tac [("c","abs h")] sums_mult 3);
+by (asm_full_simp_tac (simpset() addsimps real_mult_ac) 3);
+by (subgoal_tac "summable (%n. abs(g(h::real)(n::nat)))" 2);
+by (res_inst_tac [("g","%n. f(n::nat) * abs(h)")] summable_comparison_test 3);
+by (res_inst_tac [("x","0")] exI 3);
+by Auto_tac;
+by (res_inst_tac [("j","suminf(%n. abs(g h n))")] real_le_trans 2);
+by (auto_tac (claset() addIs [summable_rabs,summable_le],simpset() addsimps
+ [sums_summable RS suminf_mult]));
+by (auto_tac (claset() addSIs [lemma_termdiff4],simpset() addsimps
+ [(sums_summable RS suminf_mult) RS sym]));
+qed "lemma_termdiff5";
+
+(* FIXME: Long proof *)
+Goalw [deriv_def]
+ "[| summable(%n. c(n) * (K ^ n)); \
+\ summable(%n. (diffs c)(n) * (K ^ n)); \
+\ summable(%n. (diffs(diffs c))(n) * (K ^ n)); \
+\ abs(x) < abs(K) |] \
+\ ==> DERIV (%x. suminf (%n. c(n) * (x ^ n))) x :> \
+\ suminf (%n. (diffs c)(n) * (x ^ n))";
+
+by (res_inst_tac [("g","%h. suminf(%n. ((c(n) * ((x + h) ^ n)) - \
+\ (c(n) * (x ^ n))) * inverse h)")] LIM_trans 1);
+by (asm_full_simp_tac (simpset() addsimps [LIM_def]) 1);
+by (Step_tac 1);
+by (res_inst_tac [("x","abs K - abs x")] exI 1);
+by (auto_tac (claset(),simpset() addsimps [real_less_diff_eq]));
+by (dtac (abs_triangle_ineq RS order_le_less_trans) 1);
+by (res_inst_tac [("y","0")] order_le_less_trans 1);
+by Auto_tac;
+by (subgoal_tac "(%n. (c n) * (x ^ n)) sums \
+\ (suminf(%n. (c n) * (x ^ n))) & \
+\ (%n. (c n) * ((x + xa) ^ n)) sums \
+\ (suminf(%n. (c n) * ((x + xa) ^ n)))" 1);
+by (auto_tac (claset() addSIs [summable_sums],simpset()));
+by (rtac powser_inside 2 THEN rtac powser_inside 4);
+by (auto_tac (claset(),simpset() addsimps [real_add_commute]));
+by (EVERY1[rotate_tac 8, dtac sums_diff, assume_tac]);
+by (dres_inst_tac [("x","(%n. c n * (xa + x) ^ n - c n * x ^ n)"),
+ ("c","inverse xa")] sums_mult 1);
+by (rtac (sums_unique RS sym) 1);
+by (asm_full_simp_tac (simpset() addsimps [real_diff_def,
+ real_divide_def] @ real_add_ac @ real_mult_ac) 1);
+by (rtac LIM_zero_cancel 1);
+by (res_inst_tac [("g","%h. suminf (%n. c(n) * (((((x + h) ^ n) - \
+\ (x ^ n)) * inverse h) - (real n * (x ^ (n - Suc 0)))))")] LIM_trans 1);
+by (asm_full_simp_tac (simpset() addsimps [LIM_def]) 1);
+by (Step_tac 1);
+by (res_inst_tac [("x","abs K - abs x")] exI 1);
+by (auto_tac (claset(),simpset() addsimps [real_less_diff_eq]));
+by (dtac (abs_triangle_ineq RS order_le_less_trans) 1);
+by (res_inst_tac [("y","0")] order_le_less_trans 1);
+by Auto_tac;
+by (subgoal_tac "summable(%n. (diffs c)(n) * (x ^ n))" 1);
+by (rtac powser_inside 2);
+by (Auto_tac);
+by (dres_inst_tac [("c","c"),("x","x")] diffs_equiv 1);
+by (ftac sums_unique 1 THEN Auto_tac);
+by (subgoal_tac "(%n. (c n) * (x ^ n)) sums \
+\ (suminf(%n. (c n) * (x ^ n))) & \
+\ (%n. (c n) * ((x + xa) ^ n)) sums \
+\ (suminf(%n. (c n) * ((x + xa) ^ n)))" 1);
+by (Step_tac 1);
+by (auto_tac (claset() addSIs [summable_sums],simpset()));
+by (rtac powser_inside 2 THEN rtac powser_inside 4);
+by (auto_tac (claset(),simpset() addsimps [real_add_commute]));
+by (forw_inst_tac [("x","(%n. c n * (xa + x) ^ n)"),
+ ("y","(%n. c n * x ^ n)")] sums_diff 1 THEN assume_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [[sums_summable,sums_summable]
+ MRS suminf_diff,real_diff_mult_distrib2 RS sym]) 1);
+by (forw_inst_tac [("x","(%n. c n * ((xa + x) ^ n - x ^ n))"),
+ ("c","inverse xa")] sums_mult 1);
+by (asm_full_simp_tac (simpset() addsimps [sums_summable RS suminf_mult2]) 1);
+by (forw_inst_tac [("x","(%n. inverse xa * (c n * ((xa + x) ^ n - x ^ n)))"),
+ ("y","(%n. real n * c n * x ^ (n - Suc 0))")] sums_diff 1);
+by (assume_tac 1);
+by (rtac (ARITH_PROVE "z - y = x ==> - x = (y::real) - z") 1);
+by (asm_full_simp_tac (simpset() addsimps [[sums_summable,sums_summable]
+ MRS suminf_diff] @ real_add_ac @ real_mult_ac ) 1);
+by (res_inst_tac [("f","suminf")] arg_cong 1);
+by (rtac ext 1);
+by (asm_full_simp_tac (simpset() addsimps [real_diff_def,
+ real_add_mult_distrib2] @ real_add_ac @ real_mult_ac) 1);
+(* 46 *)
+by (dtac real_dense 1 THEN Step_tac 1);
+by (ftac (real_less_sum_gt_zero) 1);
+by (dres_inst_tac [("f","%n. abs(c n) * real n * \
+\ real (n - Suc 0) * (r ^ (n - 2))"),
+ ("g","%h n. c(n) * (((((x + h) ^ n) - (x ^ n)) * inverse h) - \
+\ (real n * (x ^ (n - Suc 0))))")] lemma_termdiff5 1);
+by (auto_tac (claset(),simpset() addsimps [real_add_commute]));
+by (subgoal_tac "summable(%n. abs(diffs(diffs c) n) * (r ^ n))" 1);
+by (res_inst_tac [("x","K")] powser_insidea 2 THEN Auto_tac);
+by (subgoal_tac "abs r = r" 2 THEN Auto_tac);
+by (res_inst_tac [("j1","abs x")] (real_le_trans RS abs_eqI1) 2);
+by Auto_tac;
+by (asm_full_simp_tac (simpset() addsimps [diffs_def,abs_mult,
+ real_mult_assoc RS sym]) 1);
+by (subgoal_tac "ALL n. real (Suc n) * real (Suc(Suc n)) * \
+\ abs(c(Suc(Suc n))) * (r ^ n) = diffs(diffs (%n. abs(c n))) n * (r ^ n)" 1);
+by (dres_inst_tac [("P","summable")]
+ (CLAIM "[|ALL n. f(n) = g(n); P(%n. f n)|] ==> P(%n. g(n))") 1);
+by (Auto_tac);
+by (asm_full_simp_tac (simpset() addsimps [diffs_def]) 2
+ THEN asm_full_simp_tac (simpset() addsimps [diffs_def]) 2);
+by (dtac diffs_equiv 1);
+by (dtac sums_summable 1);
+by (asm_full_simp_tac (simpset() addsimps [diffs_def] @ real_mult_ac) 1);
+by (subgoal_tac "(%n. real n * (real (Suc n) * (abs(c(Suc n)) * \
+\ (r ^ (n - Suc 0))))) = (%n. diffs(%m. real (m - Suc 0) * \
+\ abs(c m) * inverse r) n * (r ^ n))" 1);
+by (Auto_tac);
+by (rtac ext 2);
+by (asm_full_simp_tac (simpset() addsimps [diffs_def]) 2);
+by (case_tac "n" 2);
+by Auto_tac;
+(* 69 *)
+by (dtac (abs_ge_zero RS order_le_less_trans) 2);
+by (asm_full_simp_tac (simpset() addsimps real_mult_ac) 2);
+by (dtac diffs_equiv 1);
+by (dtac sums_summable 1);
+by (res_inst_tac [("a","summable (%n. real n * \
+\ (real (n - Suc 0) * abs (c n) * inverse r) * r ^ (n - Suc 0))")]
+ (CLAIM "(a = b) ==> a ==> b") 1 THEN assume_tac 2);
+by (res_inst_tac [("f","summable")] arg_cong 1 THEN rtac ext 1);
+(* 75 *)
+by (case_tac "n" 1);
+by Auto_tac;
+by (case_tac "nat" 1);
+by Auto_tac;
+by (dtac (abs_ge_zero RS order_le_less_trans) 1);
+by (auto_tac (claset(),simpset() addsimps [CLAIM_SIMP
+ "(a::real) * (b * (c * d)) = a * (b * c) * d"
+ real_mult_ac]));
+by (dtac (abs_ge_zero RS order_le_less_trans) 1);
+by (asm_full_simp_tac (simpset() addsimps [abs_mult,real_mult_assoc]) 1);
+by (rtac real_mult_le_le_mono1 1);
+by (rtac (real_add_commute RS subst) 2);
+by (simp_tac (simpset() addsimps [real_mult_assoc RS sym]) 2);
+by (rtac lemma_termdiff3 2);
+by (auto_tac (claset() addIs [(abs_triangle_ineq RS real_le_trans)],
+ simpset()));
+by (arith_tac 1);
+qed "termdiffs";
+
+(* ------------------------------------------------------------------------ *)
+(* Formal derivatives of exp, sin, and cos series *)
+(* ------------------------------------------------------------------------ *)
+
+Goalw [diffs_def]
+ "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))";
+by (rtac ext 1);
+by (rtac (fact_Suc RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (real_inverse_distrib RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym]));
+qed "exp_fdiffs";
+
+Goalw [diffs_def,real_divide_def]
+ "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 (rtac ext 1);
+by (rtac (fact_Suc RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (even_Suc RS ssubst) 1);
+by (rtac (real_inverse_distrib RS ssubst) 1);
+by Auto_tac;
+qed "sin_fdiffs";
+
+Goalw [diffs_def,real_divide_def]
+ "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 (rtac (fact_Suc RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (even_Suc RS ssubst) 1);
+by (rtac (real_inverse_distrib RS ssubst) 1);
+by Auto_tac;
+qed "sin_fdiffs2";
+
+(* thms in EvenOdd needed *)
+Goalw [diffs_def,real_divide_def]
+ "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 (rtac ext 1);
+by (rtac (fact_Suc RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (even_Suc RS ssubst) 1);
+by (rtac (real_inverse_distrib RS ssubst) 1);
+by (res_inst_tac [("z1","real (Suc n)")] (real_mult_commute RS ssubst) 1);
+by (res_inst_tac [("z1","inverse(real (Suc n))")]
+ (real_mult_commute RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc,
+ odd_not_even RS sym,odd_Suc_mult_two_ex]));
+qed "cos_fdiffs";
+
+
+Goalw [diffs_def,real_divide_def]
+ "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 (rtac (fact_Suc RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (even_Suc RS ssubst) 1);
+by (rtac (real_inverse_distrib RS ssubst) 1);
+by (res_inst_tac [("z1","real (Suc n)")] (real_mult_commute RS ssubst) 1);
+by (res_inst_tac [("z1","inverse (real (Suc n))")]
+ (real_mult_commute RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc,
+ odd_not_even RS sym,odd_Suc_mult_two_ex]));
+qed "cos_fdiffs2";
+
+(* ------------------------------------------------------------------------ *)
+(* Now at last we can get the derivatives of exp, sin and cos *)
+(* ------------------------------------------------------------------------ *)
+
+Goal "- sin x = suminf(%n. - ((if even n then 0 \
+\ else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))";
+by (auto_tac (claset() addSIs [sums_unique,sums_minus,sin_converges],
+ simpset()));
+qed "lemma_sin_minus";
+
+Goal "exp = (%x. suminf (%n. inverse (real (fact n)) * x ^ n))";
+by (auto_tac (claset() addSIs [ext],simpset() addsimps [exp_def]));
+val lemma_exp_ext = result();
+
+Goalw [exp_def] "DERIV exp x :> exp(x)";
+by (rtac (lemma_exp_ext RS ssubst) 1);
+by (subgoal_tac "DERIV (%u. suminf (%n. inverse (real (fact n)) * u ^ n)) x \
+\ :> suminf (%n. diffs (%n. inverse (real (fact n))) n * x ^ n)" 1);
+by (res_inst_tac [("K","1 + abs(x)")] termdiffs 2);
+by (auto_tac (claset() addIs [exp_converges RS sums_summable],
+ simpset() addsimps [exp_fdiffs]));
+by (arith_tac 1);
+qed "DERIV_exp";
+Addsimps [DERIV_exp];
+
+Goal "sin = (%x. suminf(%n. (if even n then 0 \
+\ else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * \
+\ x ^ n))";
+by (auto_tac (claset() addSIs [ext],simpset() addsimps [sin_def]));
+val lemma_sin_ext = result();
+
+Goal "cos = (%x. suminf(%n. (if even n then \
+\ (- 1) ^ (n div 2)/(real (fact n)) \
+\ else 0) * x ^ n))";
+by (auto_tac (claset() addSIs [ext],simpset() addsimps [cos_def]));
+val lemma_cos_ext = result();
+
+Goalw [cos_def] "DERIV sin x :> cos(x)";
+by (rtac (lemma_sin_ext RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [sin_fdiffs2 RS sym]));
+by (res_inst_tac [("K","1 + abs(x)")] termdiffs 1);
+by (auto_tac (claset() addIs [sin_converges, cos_converges, sums_summable]
+ addSIs [sums_minus RS sums_summable],
+ simpset() addsimps [cos_fdiffs,sin_fdiffs]));
+by (arith_tac 1);
+qed "DERIV_sin";
+Addsimps [DERIV_sin];
+
+Goal "DERIV cos x :> -sin(x)";
+by (rtac (lemma_cos_ext RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [lemma_sin_minus,
+ cos_fdiffs2 RS sym,real_minus_mult_eq1]));
+by (res_inst_tac [("K","1 + abs(x)")] termdiffs 1);
+by (auto_tac (claset() addIs [sin_converges,cos_converges, sums_summable]
+ addSIs [sums_minus RS sums_summable],
+ simpset() addsimps [cos_fdiffs,sin_fdiffs,diffs_minus]));
+by (arith_tac 1);
+qed "DERIV_cos";
+Addsimps [DERIV_cos];
+
+(* ------------------------------------------------------------------------ *)
+(* Properties of the exponential function *)
+(* ------------------------------------------------------------------------ *)
+
+Goalw [exp_def] "exp 0 = 1";
+by (rtac (CLAIM_SIMP "sumr 0 1 (%n. inverse (real (fact n)) * 0 ^ n) = 1"
+ [real_of_nat_one] RS subst) 1);
+by (rtac ((series_zero RS sums_unique) RS sym) 1);
+by (Step_tac 1);
+by (case_tac "m" 1);
+by (Auto_tac);
+qed "exp_zero";
+Addsimps [exp_zero];
+
+Goal "0 <= x ==> (1 + x) <= exp(x)";
+by (dtac real_le_imp_less_or_eq 1 THEN Auto_tac);
+by (rewtac exp_def);
+by (rtac real_le_trans 1);
+by (res_inst_tac [("n","2"),("f","(%n. inverse (real (fact n)) * x ^ n)")]
+ series_pos_le 2);
+by (auto_tac (claset() addIs [summable_exp],simpset()
+ addsimps [numeral_2_eq_2,realpow_ge_zero,real_0_le_mult_iff]));
+qed "exp_ge_add_one_self";
+Addsimps [exp_ge_add_one_self];
+
+Goal "0 < x ==> 1 < exp x";
+by (rtac order_less_le_trans 1);
+by (rtac exp_ge_add_one_self 2);
+by (Auto_tac);
+qed "exp_gt_one";
+Addsimps [exp_gt_one];
+
+Goal "DERIV (%x. exp (x + y)) x :> exp(x + y)";
+by (auto_tac (claset(),simpset() addsimps
+ [CLAIM_SIMP "(%x. exp (x + y)) = exp o (%x. x + y)" [ext]]));
+by (rtac (real_mult_1_right RS subst) 1);
+by (rtac DERIV_chain 1);
+by (rtac (real_add_zero_right RS subst) 2);
+by (rtac DERIV_add 2);
+by Auto_tac;
+qed "DERIV_exp_add_const";
+Addsimps [DERIV_exp_add_const];
+
+Goal "DERIV (%x. exp (-x)) x :> - exp(-x)";
+by (auto_tac (claset(),simpset() addsimps
+ [CLAIM_SIMP "(%x. exp(-x)) = exp o (%x. - x)" [ext]]));
+by (rtac (real_mult_1_right RS subst) 1);
+by (rtac (real_minus_mult_eq1 RS subst) 1);
+by (rtac (real_minus_mult_eq2 RS ssubst) 1);
+by (rtac DERIV_chain 1);
+by (rtac DERIV_minus 2);
+by Auto_tac;
+qed "DERIV_exp_minus";
+Addsimps [DERIV_exp_minus];
+
+Goal "DERIV (%x. exp (x + y) * exp (- x)) x :> 0";
+by (cut_inst_tac [("x","x"),("y2","y")] ([DERIV_exp_add_const,
+ DERIV_exp_minus] MRS DERIV_mult) 1);
+by (auto_tac (claset(),simpset() addsimps real_mult_ac));
+qed "DERIV_exp_exp_zero";
+Addsimps [DERIV_exp_exp_zero];
+
+Goal "exp(x + y)*exp(-x) = exp(y)";
+by (cut_inst_tac [("x","x"),("y2","y"),("y","0")]
+ ((CLAIM "ALL x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0") RS
+ DERIV_isconst_all) 1);
+by (Auto_tac);
+qed "exp_add_mult_minus";
+Addsimps [exp_add_mult_minus];
+
+Goal "exp(x)*exp(-x) = 1";
+by (cut_inst_tac [("x","x"),("y","0")] exp_add_mult_minus 1);
+by (Asm_full_simp_tac 1);
+qed "exp_mult_minus";
+Addsimps [exp_mult_minus];
+
+Goal "exp(-x)*exp(x) = 1";
+by (simp_tac (simpset() addsimps [real_mult_commute]) 1);
+qed "exp_mult_minus2";
+Addsimps [exp_mult_minus2];
+
+Goal "exp(-x) = inverse(exp(x))";
+by (auto_tac (claset() addIs [real_inverse_unique],simpset()));
+qed "exp_minus";
+
+Goal "exp(x + y) = exp(x) * exp(y)";
+by (cut_inst_tac [("x1","x"),("y1","y"),("z","exp x")]
+ (exp_add_mult_minus RS (CLAIM "x = y ==> z * y = z * (x::real)")) 1);
+by (asm_full_simp_tac (simpset() delsimps [exp_add_mult_minus]
+ addsimps real_mult_ac) 1);
+qed "exp_add";
+
+Goal "0 <= exp x";
+by (res_inst_tac [("t","x")] (real_sum_of_halves RS subst) 1);
+by (rtac (exp_add RS ssubst) 1 THEN Auto_tac);
+qed "exp_ge_zero";
+Addsimps [exp_ge_zero];
+
+Goal "exp x ~= 0";
+by (cut_inst_tac [("x","x")] exp_mult_minus2 1);
+by (auto_tac (claset(),simpset() delsimps [exp_mult_minus2]));
+qed "exp_not_eq_zero";
+Addsimps [exp_not_eq_zero];
+
+Goal "0 < exp x";
+by (simp_tac (simpset() addsimps
+ [CLAIM_SIMP "(x < y) = (x <= y & y ~= (x::real))" [real_le_less]]) 1);
+qed "exp_gt_zero";
+Addsimps [exp_gt_zero];
+
+Goal "0 < inverse(exp x)";
+by (auto_tac (claset() addIs [real_inverse_gt_0],simpset()));
+qed "inv_exp_gt_zero";
+Addsimps [inv_exp_gt_zero];
+
+Goal "abs(exp x) = exp x";
+by (auto_tac (claset(),simpset() addsimps [abs_eqI2]));
+qed "abs_exp_cancel";
+Addsimps [abs_exp_cancel];
+
+Goal "exp(real n * x) = exp(x) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc,
+ real_add_mult_distrib2,exp_add,real_mult_commute]));
+qed "exp_real_of_nat_mult";
+
+Goalw [real_diff_def,real_divide_def]
+ "exp(x - y) = exp(x)/(exp y)";
+by (simp_tac (simpset() addsimps [exp_add,exp_minus]) 1);
+qed "exp_diff";
+
+Goal "x < y ==> exp x < exp y";
+by (dtac ((real_less_sum_gt_zero) RS exp_gt_one) 1);
+by (multr_by_tac "inverse(exp x)" 1);
+by (auto_tac (claset(),simpset() addsimps [exp_add,exp_minus]));
+qed "exp_less_mono";
+
+Goal "exp x < exp y ==> x < y";
+by (EVERY1[rtac ccontr, dtac real_leI, dtac real_le_imp_less_or_eq]);
+by (auto_tac (claset() addDs [exp_less_mono],simpset()));
+qed "exp_less_cancel";
+
+Goal "(exp(x) < exp(y)) = (x < y)";
+by (auto_tac (claset() addIs [exp_less_mono,exp_less_cancel],simpset()));
+qed "exp_less_cancel_iff";
+AddIffs [exp_less_cancel_iff];
+
+Goalw [real_le_def] "(exp(x) <= exp(y)) = (x <= y)";
+by (Auto_tac);
+qed "exp_le_cancel_iff";
+AddIffs [exp_le_cancel_iff];
+
+Goal "(exp x = exp y) = (x = y)";
+by (auto_tac (claset(),simpset() addsimps
+ [CLAIM "(x = (y::real)) = (x <= y & y <= x)"]));
+qed "exp_inj_iff";
+AddIffs [exp_inj_iff];
+
+Goal "1 <= y ==> EX x. 0 <= x & x <= y - 1 & exp(x) = y";
+by (rtac IVT 1);
+by (auto_tac (claset() addIs [DERIV_exp RS DERIV_isCont],simpset()
+ addsimps [real_le_diff_eq]));
+by (dtac (CLAIM_SIMP "x <= y ==> (0::real) <= y - x" [real_le_diff_eq]) 1);
+by (dtac exp_ge_add_one_self 1);
+by (Asm_full_simp_tac 1);
+qed "lemma_exp_total";
+
+Goal "0 < y ==> EX x. exp x = y";
+by (res_inst_tac [("R1.0","1"),("R2.0","y")] real_linear_less2 1);
+by (dtac (order_less_imp_le RS lemma_exp_total) 1);
+by (res_inst_tac [("x","0")] exI 2);
+by (ftac real_inverse_gt_one 3);
+by (dtac (order_less_imp_le RS lemma_exp_total) 4);
+by (Step_tac 3);
+by (res_inst_tac [("x","-x")] exI 3);
+by (auto_tac (claset(),simpset() addsimps [exp_minus]));
+qed "exp_total";
+
+(* ------------------------------------------------------------------------ *)
+(* Properties of the logarithmic function *)
+(* ------------------------------------------------------------------------ *)
+
+Goalw [ln_def] "ln(exp x) = x";
+by (Simp_tac 1);
+qed "ln_exp";
+Addsimps [ln_exp];
+
+Goal "(exp(ln x) = x) = (0 < x)";
+by (auto_tac (claset() addDs [exp_total],simpset()));
+by (dtac subst 1);
+by (Auto_tac);
+qed "exp_ln_iff";
+Addsimps [exp_ln_iff];
+
+Goal "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)";
+by (rtac (exp_inj_iff RS iffD1) 1);
+by (ftac (real_mult_order) 1);
+by (auto_tac (claset(),simpset() addsimps [exp_add,exp_ln_iff RS sym]
+ delsimps [exp_inj_iff,exp_ln_iff]));
+qed "ln_mult";
+
+Goal "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)";
+by (auto_tac (claset() addSDs [(exp_ln_iff RS iffD2)],simpset()));
+qed "ln_inj_iff";
+Addsimps [ln_inj_iff];
+
+Goal "ln 1 = 0";
+by (rtac (exp_inj_iff RS iffD1) 1);
+by Auto_tac;
+qed "ln_one";
+Addsimps [ln_one];
+
+Goal "0 < x ==> ln(inverse x) = - ln x";
+by (res_inst_tac [("x1","ln x")] (real_add_left_cancel RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [real_inverse_gt_0,ln_mult RS sym]));
+qed "ln_inverse";
+
+Goalw [real_divide_def]
+ "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y";
+by (auto_tac (claset(),simpset() addsimps [real_inverse_gt_0,
+ ln_mult,ln_inverse]));
+qed "ln_div";
+
+Goal "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)";
+by (REPEAT(dtac (exp_ln_iff RS iffD2) 1));
+by (REPEAT(dtac subst 1 THEN assume_tac 2));
+by (Simp_tac 1);
+qed "ln_less_cancel_iff";
+Addsimps [ln_less_cancel_iff];
+
+Goalw [real_le_def] "[| 0 < x; 0 < y|] ==> (ln x <= ln y) = (x <= y)";
+by (Auto_tac);
+qed "ln_le_cancel_iff";
+Addsimps [ln_le_cancel_iff];
+
+Goal "0 < x ==> ln(x ^ n) = real n * ln(x)";
+by (auto_tac (claset() addSDs [exp_total],simpset()
+ addsimps [exp_real_of_nat_mult RS sym]));
+qed "ln_realpow";
+
+Goal "0 <= x ==> ln(1 + x) <= x";
+by (rtac (ln_exp RS subst) 1);
+by (rtac (ln_le_cancel_iff RS iffD2) 1);
+by Auto_tac;
+qed "ln_add_one_self_le_self";
+Addsimps [ln_add_one_self_le_self];
+
+Goal "0 < x ==> ln x < x";
+by (rtac order_less_le_trans 1);
+by (rtac ln_add_one_self_le_self 2);
+by (rtac (ln_less_cancel_iff RS iffD2) 1);
+by Auto_tac;
+qed "ln_less_self";
+Addsimps [ln_less_self];
+
+Goal "1 <= x ==> 0 <= ln x";
+by (subgoal_tac "0 < x" 1);
+by (rtac order_less_le_trans 2 THEN assume_tac 3);
+by (rtac (exp_le_cancel_iff RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps
+ [exp_ln_iff RS sym] delsimps [exp_ln_iff]));
+qed "ln_ge_zero";
+Addsimps [ln_ge_zero];
+
+Goal "1 < x ==> 0 < ln x";
+by (rtac (exp_less_cancel_iff RS iffD1) 1);
+by (rtac (exp_ln_iff RS iffD2 RS ssubst) 1);
+by Auto_tac;
+qed "ln_gt_zero";
+Addsimps [ln_gt_zero];
+
+Goal "[| 0 < x; x ~= 1 |] ==> ln x ~= 0";
+by (Step_tac 1);
+by (dtac (exp_inj_iff RS iffD2) 1);
+by (dtac (exp_ln_iff RS iffD2) 1);
+by Auto_tac;
+qed "ln_not_eq_zero";
+Addsimps [ln_not_eq_zero];
+
+Goal "[| 0 < x; x < 1 |] ==> ln x < 0";
+by (rtac (exp_less_cancel_iff RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [exp_ln_iff RS sym]
+ delsimps [exp_ln_iff]));
+qed "ln_less_zero";
+
+Goal "exp u = x ==> ln x = u";
+by Auto_tac;
+qed "exp_ln_eq";
+
+Addsimps [hypreal_less_not_refl];
+
+(* ------------------------------------------------------------------------ *)
+(* Basic properties of the trig functions *)
+(* ------------------------------------------------------------------------ *)
+
+Goalw [sin_def] "sin 0 = 0";
+by (auto_tac (claset() addSIs [(sums_unique RS sym),
+ LIMSEQ_const],simpset() addsimps [sums_def]));
+qed "sin_zero";
+Addsimps [sin_zero];
+
+Goal "(ALL m. n <= m --> f m = 0) --> f sums sumr 0 n f";
+by (auto_tac (claset() addIs [series_zero],simpset()));
+qed "lemma_series_zero2";
+
+Goalw [cos_def] "cos 0 = 1";
+by (rtac (sums_unique RS sym) 1);
+by (cut_inst_tac [("n","1"),("f","(%n. (if even n then (- 1) ^ (n div 2)/ \
+\ (real (fact n)) else 0) * 0 ^ n)")] lemma_series_zero2 1);
+by Auto_tac;
+qed "cos_zero";
+Addsimps [cos_zero];
+
+Goal "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)";
+by (rtac DERIV_mult 1 THEN Auto_tac);
+qed "DERIV_sin_sin_mult";
+Addsimps [DERIV_sin_sin_mult];
+
+Goal "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)";
+by (cut_inst_tac [("x","x")] DERIV_sin_sin_mult 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc]));
+qed "DERIV_sin_sin_mult2";
+Addsimps [DERIV_sin_sin_mult2];
+
+Goal "DERIV (%x. sin(x) ^ 2) x :> cos(x) * sin(x) + cos(x) * sin(x)";
+by (auto_tac (claset(),
+ simpset() addsimps [numeral_2_eq_2, real_mult_assoc RS sym]));
+qed "DERIV_sin_realpow2";
+Addsimps [DERIV_sin_realpow2];
+
+Goal "DERIV (%x. sin(x) ^ 2) x :> 2 * cos(x) * sin(x)";
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "DERIV_sin_realpow2a";
+Addsimps [ DERIV_sin_realpow2a];
+
+Goal "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)";
+by (rtac DERIV_mult 1 THEN Auto_tac);
+qed "DERIV_cos_cos_mult";
+Addsimps [DERIV_cos_cos_mult];
+
+Goal "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)";
+by (cut_inst_tac [("x","x")] DERIV_cos_cos_mult 1);
+by (auto_tac (claset(),simpset() addsimps real_mult_ac));
+qed "DERIV_cos_cos_mult2";
+Addsimps [DERIV_cos_cos_mult2];
+
+Goal "DERIV (%x. cos(x) ^ 2) x :> -sin(x) * cos(x) + -sin(x) * cos(x)";
+by (auto_tac (claset(),
+ simpset() addsimps [numeral_2_eq_2, real_mult_assoc RS sym]));
+qed "DERIV_cos_realpow2";
+Addsimps [DERIV_cos_realpow2];
+
+Goal "DERIV (%x. cos(x) ^ 2) x :> -2 * cos(x) * sin(x)";
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "DERIV_cos_realpow2a";
+Addsimps [DERIV_cos_realpow2a];
+
+Goal "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E";
+by (Auto_tac);
+val lemma_DERIV_subst = result();
+
+Goal "DERIV (%x. cos(x) ^ 2) x :> -(2 * cos(x) * sin(x))";
+by (rtac lemma_DERIV_subst 1);
+by (rtac DERIV_cos_realpow2a 1);
+by Auto_tac;
+qed "DERIV_cos_realpow2b";
+
+(* most useful *)
+Goal "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))";
+by (rtac lemma_DERIV_subst 1);
+by (rtac DERIV_cos_cos_mult2 1);
+by Auto_tac;
+qed "DERIV_cos_cos_mult3";
+Addsimps [DERIV_cos_cos_mult3];
+
+Goalw [real_diff_def]
+ "ALL x. DERIV (%x. sin(x) ^ 2 + cos(x) ^ 2) x :> \
+\ (2*cos(x)*sin(x) - 2*cos(x)*sin(x))";
+by (Step_tac 1);
+by (rtac DERIV_add 1);
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "DERIV_sin_circle_all";
+
+Goal "ALL x. DERIV (%x. sin(x) ^ 2 + cos(x) ^ 2) x :> 0";
+by (cut_facts_tac [DERIV_sin_circle_all] 1);
+by Auto_tac;
+qed "DERIV_sin_circle_all_zero";
+Addsimps [DERIV_sin_circle_all_zero];
+
+Goal "(sin(x) ^ 2) + (cos(x) ^ 2) = 1";
+by (cut_inst_tac [("x","x"),("y","0")]
+ (DERIV_sin_circle_all_zero RS DERIV_isconst_all) 1);
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "sin_cos_squared_add";
+Addsimps [sin_cos_squared_add];
+
+Goal "(cos(x) ^ 2) + (sin(x) ^ 2) = 1";
+by (rtac (real_add_commute RS ssubst) 1);
+by (simp_tac (simpset() delsimps [realpow_Suc]) 1);
+qed "sin_cos_squared_add2";
+Addsimps [sin_cos_squared_add2];
+
+Goal "cos x * cos x + sin x * sin x = 1";
+by (cut_inst_tac [("x","x")] sin_cos_squared_add2 1);
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "sin_cos_squared_add3";
+Addsimps [sin_cos_squared_add3];
+
+Goal "(sin(x) ^ 2) = 1 - (cos(x) ^ 2)";
+by (res_inst_tac [("x1","(cos(x) ^ 2)")] (real_add_right_cancel RS iffD1) 1);
+by (simp_tac (simpset() delsimps [realpow_Suc]) 1);
+qed "sin_squared_eq";
+
+Goal "(cos(x) ^ 2) = 1 - (sin(x) ^ 2)";
+by (res_inst_tac [("x1","(sin(x) ^ 2)")] (real_add_right_cancel RS iffD1) 1);
+by (simp_tac (simpset() delsimps [realpow_Suc]) 1);
+qed "cos_squared_eq";
+
+Goal "[| 1 < x; 0 <= y |] ==> 1 < x + (y::real)";
+by (arith_tac 1);
+qed "real_gt_one_ge_zero_add_less";
+
+Goalw [real_le_def] "abs(sin x) <= 1";
+by (rtac notI 1);
+by (dtac realpow_two_gt_one 1);
+by (auto_tac (claset(),simpset() delsimps [realpow_Suc]));
+by (dres_inst_tac [("r1","cos x")] (realpow_two_le RSN
+ (2, real_gt_one_ge_zero_add_less)) 1);
+by (asm_full_simp_tac (simpset() addsimps [numeral_2_eq_2 RS sym]
+ delsimps [realpow_Suc]) 1);
+qed "abs_sin_le_one";
+Addsimps [abs_sin_le_one];
+
+Goal "- 1 <= sin x";
+by (full_simp_tac (simpset() addsimps [simplify (simpset()) (abs_sin_le_one RS
+ (abs_le_interval_iff RS iffD1))]) 1);
+qed "sin_ge_minus_one";
+Addsimps [sin_ge_minus_one];
+
+Goal "-1 <= sin x";
+by (rtac (simplify (simpset()) sin_ge_minus_one) 1);
+qed "sin_ge_minus_one2";
+Addsimps [sin_ge_minus_one2];
+
+Goal "sin x <= 1";
+by (full_simp_tac (simpset() addsimps [abs_sin_le_one RS
+ (abs_le_interval_iff RS iffD1)]) 1);
+qed "sin_le_one";
+Addsimps [sin_le_one];
+
+Goalw [real_le_def] "abs(cos x) <= 1";
+by (rtac notI 1);
+by (dtac realpow_two_gt_one 1);
+by (auto_tac (claset(),simpset() delsimps [realpow_Suc]));
+by (dres_inst_tac [("r1","sin x")] (realpow_two_le RSN
+ (2, real_gt_one_ge_zero_add_less)) 1);
+by (asm_full_simp_tac (simpset() addsimps [numeral_2_eq_2 RS sym]
+ delsimps [realpow_Suc]) 1);
+qed "abs_cos_le_one";
+Addsimps [abs_cos_le_one];
+
+Goal "- 1 <= cos x";
+by (full_simp_tac (simpset() addsimps [simplify (simpset())(abs_cos_le_one RS
+ (abs_le_interval_iff RS iffD1))]) 1);
+qed "cos_ge_minus_one";
+Addsimps [cos_ge_minus_one];
+
+Goal "-1 <= cos x";
+by (rtac (simplify (simpset()) cos_ge_minus_one) 1);
+qed "cos_ge_minus_one2";
+Addsimps [cos_ge_minus_one2];
+
+Goal "cos x <= 1";
+by (full_simp_tac (simpset() addsimps [abs_cos_le_one RS
+ (abs_le_interval_iff RS iffD1)]) 1);
+qed "cos_le_one";
+Addsimps [cos_le_one];
+
+Goal "DERIV g x :> m ==> \
+\ DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m";
+by (rtac lemma_DERIV_subst 1);
+by (res_inst_tac [("f","(%x. x ^ n)")] DERIV_chain2 1);
+by (rtac DERIV_pow 1 THEN Auto_tac);
+qed "DERIV_fun_pow";
+
+Goal "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m";
+by (rtac lemma_DERIV_subst 1);
+by (res_inst_tac [("f","exp")] DERIV_chain2 1);
+by (rtac DERIV_exp 1 THEN Auto_tac);
+qed "DERIV_fun_exp";
+
+Goal "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m";
+by (rtac lemma_DERIV_subst 1);
+by (res_inst_tac [("f","sin")] DERIV_chain2 1);
+by (rtac DERIV_sin 1 THEN Auto_tac);
+qed "DERIV_fun_sin";
+
+Goal "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m";
+by (rtac lemma_DERIV_subst 1);
+by (res_inst_tac [("f","cos")] DERIV_chain2 1);
+by (rtac DERIV_cos 1 THEN Auto_tac);
+qed "DERIV_fun_cos";
+
+(* FIXME: remove this quick, crude tactic *)
+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];
+
+
+fun deriv_tac i = (resolve_tac deriv_rulesI i) ORELSE
+ ((rtac (read_instantiate [("f",get_fun_name (getgoal i))]
+ DERIV_chain2) i) handle DERIV_name => no_tac);
+
+val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));
+
+(* lemma *)
+Goal "ALL 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";
+by (Step_tac 1 THEN rtac lemma_DERIV_subst 1);
+by DERIV_tac;
+by (auto_tac (claset(),simpset() addsimps [real_diff_def,
+ real_add_mult_distrib,real_add_mult_distrib2] @
+ real_mult_ac @ real_add_ac));
+val lemma_DERIV_sin_cos_add = result();
+
+Goal "(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";
+by (cut_inst_tac [("y","0"),("x","x"),("y7","y")]
+ (lemma_DERIV_sin_cos_add RS DERIV_isconst_all) 1);
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "sin_cos_add";
+Addsimps [sin_cos_add];
+
+Goal "sin (x + y) = sin x * cos y + cos x * sin y";
+by (cut_inst_tac [("x","x"),("y","y")] sin_cos_add 1);
+by (auto_tac (claset() addSDs [real_sum_squares_cancel_a],
+ simpset() addsimps [numeral_2_eq_2] delsimps [sin_cos_add]));
+qed "sin_add";
+
+Goal "cos (x + y) = cos x * cos y - sin x * sin y";
+by (cut_inst_tac [("x","x"),("y","y")] sin_cos_add 1);
+by (auto_tac (claset() addSDs [real_sum_squares_cancel_a],simpset() addsimps [numeral_2_eq_2] delsimps [sin_cos_add]));
+qed "cos_add";
+
+Goal "ALL x. \
+\ DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0";
+by (Step_tac 1 THEN rtac lemma_DERIV_subst 1);
+by DERIV_tac;
+by (auto_tac (claset(),simpset() addsimps [real_diff_def,
+ real_add_mult_distrib,real_add_mult_distrib2]
+ @ real_mult_ac @ real_add_ac));
+val lemma_DERIV_sin_cos_minus = result();
+
+Goal "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0";
+by (cut_inst_tac [("y","0"),("x","x")]
+ (lemma_DERIV_sin_cos_minus RS DERIV_isconst_all) 1);
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "sin_cos_minus";
+Addsimps [sin_cos_minus];
+
+Goal "sin (-x) = -sin(x)";
+by (cut_inst_tac [("x","x")] sin_cos_minus 1);
+by (auto_tac (claset() addSDs [real_sum_squares_cancel_a],
+ simpset() addsimps [numeral_2_eq_2] delsimps [sin_cos_minus]));
+qed "sin_minus";
+Addsimps [sin_minus];
+
+Goal "cos (-x) = cos(x)";
+by (cut_inst_tac [("x","x")] sin_cos_minus 1);
+by (auto_tac (claset() addSDs [real_sum_squares_cancel_a],
+ simpset() addsimps [numeral_2_eq_2] delsimps [sin_cos_minus]));
+qed "cos_minus";
+Addsimps [cos_minus];
+
+Goalw [real_diff_def] "sin (x - y) = sin x * cos y - cos x * sin y";
+by (simp_tac (simpset() addsimps [sin_add]) 1);
+qed "sin_diff";
+
+Goal "sin (x - y) = cos y * sin x - sin y * cos x";
+by (simp_tac (simpset() addsimps [sin_diff,real_mult_commute]) 1);
+qed "sin_diff2";
+
+Goalw [real_diff_def] "cos (x - y) = cos x * cos y + sin x * sin y";
+by (simp_tac (simpset() addsimps [cos_add]) 1);
+qed "cos_diff";
+
+Goal "cos (x - y) = cos y * cos x + sin y * sin x";
+by (simp_tac (simpset() addsimps [cos_diff,real_mult_commute]) 1);
+qed "cos_diff2";
+
+Goal "sin(2 * x) = 2* sin x * cos x";
+by (cut_inst_tac [("x","x"),("y","x")] sin_add 1);
+by Auto_tac;
+qed "sin_double";
+
+Addsimps [sin_double];
+
+Goal "cos(2* x) = (cos(x) ^ 2) - (sin(x) ^ 2)";
+by (cut_inst_tac [("x","x"),("y","x")] cos_add 1);
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "cos_double";
+
+(* ------------------------------------------------------------------------ *)
+(* Show that there's a least positive x with cos(x) = 0; hence define pi *)
+(* ------------------------------------------------------------------------ *)
+
+Goal "(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * \
+\ x ^ (2 * n + 1)) sums sin x";
+by (cut_inst_tac [("x2","x")] (CLAIM "0 < (2::nat)" RS ((sin_converges
+ RS sums_summable) RS sums_group)) 1);
+by (auto_tac (claset(),simpset() addsimps mult_ac@[sin_def]));
+qed "sin_paired";
+
+Goal "real (Suc (Suc (Suc (Suc 2)))) = \
+\ real (2::nat) * real (Suc 2)";
+by (simp_tac (simpset() addsimps [numeral_2_eq_2, real_of_nat_Suc]) 1);
+val lemma_real_of_nat_six_mult = result();
+
+Goal "[|0 < x; x < 2 |] ==> 0 < sin x";
+by (cut_inst_tac [("x2","x")] (CLAIM "0 < (2::nat)" RS ((sin_paired
+ RS sums_summable) RS sums_group)) 1);
+by (rotate_tac 2 1);
+by (dtac ((sin_paired RS sums_unique) RS ssubst) 1);
+by (auto_tac (claset(),simpset() delsimps [fact_Suc,realpow_Suc]));
+by (ftac sums_unique 1);
+by (auto_tac (claset(),simpset() delsimps [fact_Suc,realpow_Suc]));
+by (res_inst_tac [("n1","0")] (series_pos_less RSN (2,order_le_less_trans)) 1);
+by (auto_tac (claset(),simpset() delsimps [fact_Suc,realpow_Suc]));
+by (etac sums_summable 1);
+by (case_tac "m=0" 1);
+by (Asm_simp_tac 1);
+by (res_inst_tac [("z","real (Suc (Suc (Suc (Suc 2))))")]
+ (CLAIM "[|(0::real)<z; z*x<z*y |] ==> x<y") 1);
+by (asm_simp_tac (simpset() addsimps [numeral_2_eq_2 RS sym, real_mult_assoc RS sym]) 2);
+by (stac (CLAIM "6 = 2 * (3::real)") 2);
+by (rtac real_mult_less_mono 2);
+by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc] delsimps [fact_Suc]));
+by (rtac (fact_Suc RS ssubst) 1);
+by (rtac (fact_Suc RS ssubst) 1);
+by (rtac (fact_Suc RS ssubst) 1);
+by (rtac (fact_Suc RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (simp_tac (simpset() addsimps [real_divide_def,
+ real_inverse_distrib] delsimps [fact_Suc]) 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym]
+ delsimps [fact_Suc]));
+by (multr_by_tac "real (Suc (Suc (4*m)))" 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc]
+ delsimps [fact_Suc]));
+by (multr_by_tac "real (Suc (Suc (Suc (4*m))))" 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc]
+ delsimps [fact_Suc]));
+by (auto_tac (claset(),simpset() addsimps [CLAIM
+ "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * (x::real))"]
+ delsimps [fact_Suc]));
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc,realpow_gt_zero]
+ delsimps [fact_Suc]));
+by (rtac real_mult_less_mono 1);
+by (ALLGOALS(Asm_simp_tac));
+by (TRYALL(rtac real_less_trans));
+by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc] delsimps [fact_Suc]));
+by (res_inst_tac [("y","0")] order_less_le_trans 1);
+by (ALLGOALS(Asm_simp_tac));
+qed "sin_gt_zero";
+
+Goal "[|0 < x; x < 2 |] ==> 0 < sin x";
+by (auto_tac (claset() addIs [sin_gt_zero],simpset()));
+qed "sin_gt_zero1";
+
+Goal "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1";
+by (auto_tac (claset(),simpset() addsimps [cos_squared_eq,
+ real_minus_add_distrib RS sym, real_minus_zero_less_iff2,sin_gt_zero1,
+ real_add_order,realpow_gt_zero,cos_double] delsimps
+ [realpow_Suc,real_minus_add_distrib]));
+qed "cos_double_less_one";
+
+Goal "(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) \
+\ sums cos x";
+by (cut_inst_tac [("x2","x")] (CLAIM "0 < (2::nat)" RS ((cos_converges
+ RS sums_summable) RS sums_group)) 1);
+by (auto_tac (claset(),simpset() addsimps mult_ac@[cos_def]));
+qed "cos_paired";
+
+Addsimps [realpow_gt_zero];
+
+Goal "cos (2) < 0";
+by (cut_inst_tac [("x","2")] cos_paired 1);
+by (dtac sums_minus 1);
+by (rtac (CLAIM "- x < -y ==> (y::real) < x") 1);
+by (ftac sums_unique 1 THEN Auto_tac);
+by (res_inst_tac [("R2.0",
+ "sumr 0 (Suc (Suc (Suc 0))) (%n. -((- 1) ^ n /(real (fact(2 * n))) \
+\ * 2 ^ (2 * n)))")] real_less_trans 1);
+by (simp_tac (simpset() addsimps [fact_num_eq_if,realpow_num_eq_if]
+ delsimps [fact_Suc,realpow_Suc]) 1);
+by (simp_tac (simpset() addsimps [real_mult_assoc]
+ delsimps [sumr_Suc]) 1);
+by (rtac sumr_pos_lt_pair 1);
+by (etac sums_summable 1);
+by (Step_tac 1);
+by (simp_tac (simpset() addsimps [real_divide_def,real_mult_assoc RS sym]
+ delsimps [fact_Suc]) 1);
+by (rtac real_mult_inverse_cancel2 1);
+by (TRYALL(rtac (real_of_nat_fact_gt_zero)));
+by (simp_tac (simpset() addsimps [real_mult_assoc RS sym]
+ delsimps [fact_Suc]) 1);
+by (rtac ((CLAIM "real(n::nat) * 4 = real(4 * n)") RS ssubst) 1);
+by (rtac (fact_Suc RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac (real_of_nat_mult RS ssubst) 1);
+by (rtac real_mult_less_mono 1);
+by (Force_tac 1);
+by (Force_tac 2);
+by (rtac real_of_nat_fact_gt_zero 2);
+by (rtac (real_of_nat_less_iff RS iffD2) 1);
+by (rtac fact_less_mono 1);
+by Auto_tac;
+qed "cos_two_less_zero";
+Addsimps [cos_two_less_zero];
+Addsimps [cos_two_less_zero RS real_not_refl2];
+Addsimps [cos_two_less_zero RS order_less_imp_le];
+
+Goal "EX! x. 0 <= x & x <= 2 & cos x = 0";
+by (subgoal_tac "EX x. 0 <= x & x <= 2 & cos x = 0" 1);
+by (rtac IVT2 2);
+by (auto_tac (claset() addIs [DERIV_isCont,DERIV_cos],simpset ()));
+by (cut_inst_tac [("R1.0","xa"),("R2.0","y")] real_linear 1);
+by (rtac ccontr 1);
+by (subgoal_tac "(ALL x. cos differentiable x) & \
+\ (ALL x. isCont cos x)" 1);
+by (auto_tac (claset() addIs [DERIV_cos,DERIV_isCont],simpset()
+ addsimps [differentiable_def]));
+by (dres_inst_tac [("f","cos")] Rolle 1);
+by (dres_inst_tac [("f","cos")] Rolle 5);
+by (auto_tac (claset() addSDs [DERIV_cos RS DERIV_unique],
+ simpset() addsimps [differentiable_def]));
+by (dres_inst_tac [("y1","xa")] (order_le_less_trans RS sin_gt_zero) 1);
+by (assume_tac 1 THEN rtac order_less_le_trans 1);
+by (dres_inst_tac [("y1","y")] (order_le_less_trans RS sin_gt_zero) 4);
+by (assume_tac 4 THEN rtac order_less_le_trans 4);
+by (assume_tac 1 THEN assume_tac 3);
+by (ALLGOALS (Asm_full_simp_tac));
+qed "cos_is_zero";
+
+Goalw [pi_def] "pi/2 = (@x. 0 <= x & x <= 2 & cos x = 0)";
+by Auto_tac;
+qed "pi_half";
+
+Goal "cos (pi / 2) = 0";
+by (rtac (cos_is_zero RS ex1E) 1);
+by (auto_tac (claset() addSIs [someI2],
+ simpset() addsimps [pi_half]));
+qed "cos_pi_half";
+Addsimps [cos_pi_half];
+
+Goal "0 < pi / 2";
+by (rtac (cos_is_zero RS ex1E) 1);
+by (auto_tac (claset(),simpset() addsimps [pi_half]));
+by (rtac someI2 1);
+by (Blast_tac 1);
+by (Step_tac 1);
+by (dres_inst_tac [("y","xa")] real_le_imp_less_or_eq 1);
+by (Step_tac 1 THEN Asm_full_simp_tac 1);
+qed "pi_half_gt_zero";
+Addsimps [pi_half_gt_zero];
+Addsimps [(pi_half_gt_zero RS real_not_refl2) RS not_sym];
+Addsimps [pi_half_gt_zero RS order_less_imp_le];
+
+Goal "pi / 2 < 2";
+by (rtac (cos_is_zero RS ex1E) 1);
+by (auto_tac (claset(),simpset() addsimps [pi_half]));
+by (rtac someI2 1);
+by (Blast_tac 1);
+by (Step_tac 1);
+by (dres_inst_tac [("x","xa")] order_le_imp_less_or_eq 1);
+by (Step_tac 1 THEN Asm_full_simp_tac 1);
+qed "pi_half_less_two";
+Addsimps [pi_half_less_two];
+Addsimps [pi_half_less_two RS real_not_refl2];
+Addsimps [pi_half_less_two RS order_less_imp_le];
+
+Goal "0 < pi";
+by (multr_by_tac "inverse 2" 1);
+by Auto_tac;
+qed "pi_gt_zero";
+Addsimps [pi_gt_zero];
+Addsimps [(pi_gt_zero RS real_not_refl2) RS not_sym];
+Addsimps [pi_gt_zero RS CLAIM "(x::real) < y ==> ~ y < x"];
+
+Goal "0 <= pi";
+by (auto_tac (claset() addIs [order_less_imp_le],simpset()));
+qed "pi_ge_zero";
+Addsimps [pi_ge_zero];
+
+Goal "-(pi/2) < 0";
+by Auto_tac;
+qed "minus_pi_half_less_zero";
+Addsimps [minus_pi_half_less_zero];
+
+Goal "sin(pi/2) = 1";
+by (cut_inst_tac [("x","pi/2")] sin_cos_squared_add2 1);
+by (cut_facts_tac [[pi_half_gt_zero,pi_half_less_two] MRS sin_gt_zero] 1);
+by (auto_tac (claset(), simpset() addsimps [numeral_2_eq_2]));
+qed "sin_pi_half";
+Addsimps [sin_pi_half];
+
+Goal "cos pi = - 1";
+by (cut_inst_tac [("x","pi/2"),("y","pi/2")] cos_add 1);
+by (Asm_full_simp_tac 1);
+qed "cos_pi";
+Addsimps [cos_pi];
+
+Goal "sin pi = 0";
+by (cut_inst_tac [("x","pi/2"),("y","pi/2")] sin_add 1);
+by (Asm_full_simp_tac 1);
+qed "sin_pi";
+Addsimps [sin_pi];
+
+Goalw [real_diff_def] "sin x = cos (pi/2 - x)";
+by (simp_tac (simpset() addsimps [cos_add]) 1);
+qed "sin_cos_eq";
+
+Goal "-sin x = cos (x + pi/2)";
+by (simp_tac (simpset() addsimps [cos_add]) 1);
+qed "minus_sin_cos_eq";
+Addsimps [minus_sin_cos_eq RS sym];
+
+Goalw [real_diff_def] "cos x = sin (pi/2 - x)";
+by (simp_tac (simpset() addsimps [sin_add]) 1);
+qed "cos_sin_eq";
+Addsimps [sin_cos_eq RS sym, cos_sin_eq RS sym];
+
+Goal "sin (x + pi) = - sin x";
+by (simp_tac (simpset() addsimps [sin_add]) 1);
+qed "sin_periodic_pi";
+Addsimps [sin_periodic_pi];
+
+Goal "sin (pi + x) = - sin x";
+by (simp_tac (simpset() addsimps [sin_add]) 1);
+qed "sin_periodic_pi2";
+Addsimps [sin_periodic_pi2];
+
+Goal "cos (x + pi) = - cos x";
+by (simp_tac (simpset() addsimps [cos_add]) 1);
+qed "cos_periodic_pi";
+Addsimps [cos_periodic_pi];
+
+Goal "sin (x + 2*pi) = sin x";
+by (simp_tac (simpset() addsimps [sin_add,cos_double,numeral_2_eq_2]) 1);
+ (*FIXME: just needs x^n for literals!*)
+qed "sin_periodic";
+Addsimps [sin_periodic];
+
+Goal "cos (x + 2*pi) = cos x";
+by (simp_tac (simpset() addsimps [cos_add,cos_double,numeral_2_eq_2]) 1);
+ (*FIXME: just needs x^n for literals!*)
+qed "cos_periodic";
+Addsimps [cos_periodic];
+
+Goal "cos (real n * pi) = (-(1::real)) ^ n";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps
+ [real_of_nat_Suc,real_add_mult_distrib]));
+qed "cos_npi";
+Addsimps [cos_npi];
+
+Goal "sin (real (n::nat) * pi) = 0";
+by (induct_tac "n" 1);
+by (auto_tac (claset(),simpset() addsimps
+ [real_of_nat_Suc,real_add_mult_distrib]));
+qed "sin_npi";
+Addsimps [sin_npi];
+
+Goal "sin (pi * real (n::nat)) = 0";
+by (cut_inst_tac [("n","n")] sin_npi 1);
+by (auto_tac (claset(),simpset() addsimps [real_mult_commute]
+ delsimps [sin_npi]));
+qed "sin_npi2";
+Addsimps [sin_npi2];
+
+Goal "cos (2 * pi) = 1";
+by (simp_tac (simpset() addsimps [cos_double,numeral_2_eq_2]) 1);
+ (*FIXME: just needs x^n for literals!*)
+qed "cos_two_pi";
+Addsimps [cos_two_pi];
+
+Goal "sin (2 * pi) = 0";
+by (Simp_tac 1);
+qed "sin_two_pi";
+Addsimps [sin_two_pi];
+
+Goal "[| 0 < x; x < pi/2 |] ==> 0 < sin x";
+by (rtac sin_gt_zero 1);
+by (rtac real_less_trans 2 THEN assume_tac 2);
+by Auto_tac;
+qed "sin_gt_zero2";
+
+Goal "[| - pi/2 < x; x < 0 |] ==> sin x < 0";
+by (rtac (CLAIM "(0::real) < - x ==> x < 0") 1);
+by (rtac (sin_minus RS subst) 1);
+by (rtac sin_gt_zero2 1);
+by (rtac (CLAIM "-y < x ==> -x < (y::real)") 2);
+by Auto_tac;
+qed "sin_less_zero";
+
+Goal "[| 0 < x; x < pi/2 |] ==> 0 < cos x";
+by (cut_inst_tac [("f","cos"),("a","0"),("b","x"),("y","0")] IVT2_objl 1);
+by (Step_tac 1);
+by (cut_facts_tac [cos_is_zero] 5);
+by (Step_tac 5);
+by (dres_inst_tac [("x","xa")] spec 5);
+by (dres_inst_tac [("x","pi/2")] spec 5);
+by (auto_tac (claset() addSDs [ pi_half_less_two RS order_less_trans,
+ CLAIM "~ m <= n ==> n < (m::real)"]
+ addIs [DERIV_isCont,DERIV_cos],simpset()));
+qed "cos_gt_zero";
+
+Goal "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x";
+by (res_inst_tac [("R1.0","x"),("R2.0","0")] real_linear_less2 1);
+by (rtac (cos_minus RS subst) 1);
+by (rtac cos_gt_zero 1);
+by (rtac (CLAIM "-y < x ==> -x < (y::real)") 2);
+by (auto_tac (claset() addIs [cos_gt_zero],simpset()));
+qed "cos_gt_zero_pi";
+
+Goal "[| -(pi/2) <= x; x <= pi/2 |] ==> 0 <= cos x";
+by (auto_tac (claset(),HOL_ss addsimps [real_le_less,
+ cos_gt_zero_pi]));
+by Auto_tac;
+qed "cos_ge_zero";
+
+Goal "[| 0 < x; x < pi |] ==> 0 < sin x";
+by (rtac (sin_cos_eq RS ssubst) 1);
+by (rotate_tac 1 1);
+by (dtac (real_sum_of_halves RS ssubst) 1);
+by (auto_tac (claset() addSIs [cos_gt_zero_pi],
+ simpset() delsimps [sin_cos_eq RS sym]));
+qed "sin_gt_zero_pi";
+
+Goal "[| 0 <= x; x <= pi |] ==> 0 <= sin x";
+by (auto_tac (claset(),simpset() addsimps [real_le_less,
+ sin_gt_zero_pi]));
+qed "sin_ge_zero";
+
+Goal "[| - 1 <= y; y <= 1 |] ==> EX! x. 0 <= x & x <= pi & (cos x = y)";
+by (subgoal_tac "EX x. 0 <= x & x <= pi & cos x = y" 1);
+by (rtac IVT2 2);
+by (auto_tac (claset() addIs [order_less_imp_le,DERIV_isCont,DERIV_cos],
+ simpset ()));
+by (cut_inst_tac [("R1.0","xa"),("R2.0","y")] real_linear 1);
+by (rtac ccontr 1 THEN Auto_tac);
+by (dres_inst_tac [("f","cos")] Rolle 1);
+by (dres_inst_tac [("f","cos")] Rolle 5);
+by (auto_tac (claset() addIs [order_less_imp_le,DERIV_isCont,DERIV_cos]
+ addSDs [DERIV_cos RS DERIV_unique],simpset() addsimps [differentiable_def]));
+by (auto_tac (claset() addDs [[order_le_less_trans,order_less_le_trans] MRS
+ sin_gt_zero_pi],simpset()));
+qed "cos_total";
+
+Goal "[| - 1 <= y; y <= 1 |] ==> \
+\ EX! x. -(pi/2) <= x & x <= pi/2 & (sin x = y)";
+by (rtac ccontr 1);
+by (subgoal_tac "ALL x. (-(pi/2) <= x & x <= pi/2 & (sin x = y)) \
+\ = (0 <= (x + pi/2) & (x + pi/2) <= pi & \
+\ (cos(x + pi/2) = -y))" 1);
+by (etac swap 1);
+by (asm_full_simp_tac (simpset() delsimps [minus_sin_cos_eq RS sym]) 1);
+by (dtac (CLAIM "(x::real) <= y ==> -y <= -x") 1);
+by (dtac (CLAIM "(x::real) <= y ==> -y <= -x") 1);
+by (dtac cos_total 1);
+by (Asm_full_simp_tac 1);
+by (etac ex1E 1);
+by (res_inst_tac [("a","x - (pi/2)")] ex1I 1);
+by (simp_tac (simpset() addsimps [real_add_assoc]) 1);
+by (rotate_tac 3 1);
+by (dres_inst_tac [("x","xa + pi/2")] spec 1);
+by (Step_tac 1);
+by (TRYALL(Asm_full_simp_tac));
+by (auto_tac (claset(),simpset() addsimps [CLAIM "(-x <= y) = (-y <= (x::real))"]));
+qed "sin_total";
+
+Goal "(EX n. P (n::nat)) = (EX n. P n & (ALL m. m < n --> ~ P m))";
+by (rtac iffI 1);
+by (rtac contrapos_pp 1 THEN assume_tac 1);
+by (EVERY1[Simp_tac, rtac allI, rtac nat_less_induct]);
+by (Auto_tac);
+qed "less_induct_ex_iff";
+
+Goal "[| 0 < y; 0 <= x |] ==> \
+\ EX n. real n * y <= x & x < real (Suc n) * y";
+by (auto_tac (claset() addSDs [reals_Archimedean3],simpset()));
+by (dres_inst_tac [("x","x")] spec 1);
+by (dtac (less_induct_ex_iff RS iffD1) 1 THEN Step_tac 1);
+by (case_tac "n" 1);
+by (res_inst_tac [("x","nat")] exI 2);
+by Auto_tac;
+qed "reals_Archimedean4";
+
+(* Pre Isabelle99-2 proof was simpler- numerals arithmetic
+ now causes some unwanted re-arrangements of literals! *)
+Goal "[| 0 <= x; cos x = 0 |] ==> \
+\ EX n. ~even n & x = real n * (pi/2)";
+by (dtac (pi_gt_zero RS reals_Archimedean4) 1);
+by (Step_tac 1);
+by (subgoal_tac
+ "0 <= x - real n * pi & (x - real n * pi) <= pi & \
+\ (cos(x - real n * pi) = 0)" 1);
+by (Step_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [real_of_nat_Suc,
+ real_add_mult_distrib]) 2);
+by (asm_full_simp_tac (simpset() addsimps [cos_diff]) 1);
+by (asm_full_simp_tac (simpset() addsimps [cos_diff]) 2);
+by (subgoal_tac "EX! x. 0 <= x & x <= pi & cos x = 0" 1);
+by (rtac cos_total 2);
+by (Step_tac 1);
+by (dres_inst_tac [("x","x - real n * pi")] spec 1);
+by (dres_inst_tac [("x","pi/2")] spec 1);
+by (asm_full_simp_tac (simpset() addsimps [cos_diff]) 1);
+by (res_inst_tac [("x","Suc (2 * n)")] exI 1);
+by (asm_full_simp_tac (simpset() addsimps [real_of_nat_Suc,
+ real_add_mult_distrib]) 1);
+by Auto_tac;
+qed "cos_zero_lemma";
+
+Goal "[| 0 <= x; sin x = 0 |] ==> \
+\ EX n. even n & x = real n * (pi/2)";
+by (subgoal_tac
+ "EX n. ~ even n & x + pi/2 = real n * (pi/2)" 1);
+by (rtac cos_zero_lemma 2);
+by (Step_tac 1);
+by (res_inst_tac [("x","n - 1")] exI 1);
+by (rtac (CLAIM "-y <= x ==> -x <= (y::real)") 2);
+by (rtac real_le_trans 2 THEN assume_tac 3);
+by (auto_tac (claset(),simpset() addsimps [odd_not_even RS sym,
+ odd_Suc_mult_two_ex,real_of_nat_Suc,
+ real_add_mult_distrib,real_mult_assoc RS sym]));
+qed "sin_zero_lemma";
+
+(* also spoilt by numeral arithmetic *)
+Goal "(cos x = 0) = \
+\ ((EX n. ~even n & (x = real n * (pi/2))) | \
+\ (EX n. ~even n & (x = -(real n * (pi/2)))))";
+by (rtac iffI 1);
+by (cut_inst_tac [("x","x")] (CLAIM "0 <= (x::real) | x <= 0") 1);
+by (Step_tac 1);
+by (dtac cos_zero_lemma 1);
+by (dtac (CLAIM "(x::real) <= 0 ==> 0 <= -x") 3);
+by (dtac cos_zero_lemma 3);
+by (Step_tac 1);
+by (dtac (CLAIM "-x = y ==> x = -(y::real)") 2);
+by (auto_tac (claset(),HOL_ss addsimps [odd_not_even RS sym,
+ odd_Suc_mult_two_ex,real_of_nat_Suc,real_add_mult_distrib]));
+by (auto_tac (claset(),simpset() addsimps [cos_add]));
+qed "cos_zero_iff";
+
+(* ditto: but to a lesser extent *)
+Goal "(sin x = 0) = \
+\ ((EX n. even n & (x = real n * (pi/2))) | \
+\ (EX n. even n & (x = -(real n * (pi/2)))))";
+by (rtac iffI 1);
+by (cut_inst_tac [("x","x")] (CLAIM "0 <= (x::real) | x <= 0") 1);
+by (Step_tac 1);
+by (dtac sin_zero_lemma 1);
+by (dtac (CLAIM "(x::real) <= 0 ==> 0 <= -x") 3);
+by (dtac sin_zero_lemma 3);
+by (Step_tac 1);
+by (dtac (CLAIM "-x = y ==> x = -(y::real)") 2);
+by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex]));
+qed "sin_zero_iff";
+
+(* ------------------------------------------------------------------------ *)
+(* Tangent *)
+(* ------------------------------------------------------------------------ *)
+
+Goalw [tan_def] "tan 0 = 0";
+by (Simp_tac 1);
+qed "tan_zero";
+Addsimps [tan_zero];
+
+Goalw [tan_def] "tan pi = 0";
+by (Simp_tac 1);
+qed "tan_pi";
+Addsimps [tan_pi];
+
+Goalw [tan_def] "tan (real (n::nat) * pi) = 0";
+by (Simp_tac 1);
+qed "tan_npi";
+Addsimps [tan_npi];
+
+Goalw [tan_def] "tan (-x) = - tan x";
+by (simp_tac (simpset() addsimps [real_minus_mult_eq1]) 1);
+qed "tan_minus";
+Addsimps [tan_minus];
+
+Goalw [tan_def] "tan (x + 2*pi) = tan x";
+by (Simp_tac 1);
+qed "tan_periodic";
+Addsimps [tan_periodic];
+
+Goalw [tan_def,real_divide_def]
+ "[| cos x ~= 0; cos y ~= 0 |] \
+\ ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)";
+by (auto_tac (claset(),simpset() addsimps [real_inverse_distrib RS sym]
+ @ real_mult_ac));
+by (res_inst_tac [("c1","cos x * cos y")] (real_mult_right_cancel RS subst) 1);
+by (auto_tac (claset(), simpset() addsimps [real_mult_assoc,
+ real_mult_not_zero,real_diff_mult_distrib,cos_add]));
+val lemma_tan_add1 = result();
+Addsimps [lemma_tan_add1];
+
+Goalw [tan_def]
+ "[| cos x ~= 0; cos y ~= 0 |] \
+\ ==> tan x + tan y = sin(x + y)/(cos x * cos y)";
+by (res_inst_tac [("c1","cos x * cos y")] (real_mult_right_cancel RS subst) 1);
+by (auto_tac (claset(), simpset() addsimps [real_mult_assoc,
+ real_mult_not_zero,real_add_mult_distrib]));
+by (simp_tac (simpset() addsimps [sin_add]) 1);
+qed "add_tan_eq";
+
+Goal "[| cos x ~= 0; cos y ~= 0; cos (x + y) ~= 0 |] \
+\ ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))";
+by (asm_simp_tac (simpset() addsimps [add_tan_eq]) 1);
+by (simp_tac (simpset() addsimps [tan_def]) 1);
+qed "tan_add";
+
+Goal "[| cos x ~= 0; cos (2 * x) ~= 0 |] \
+\ ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))";
+by (auto_tac (claset(),simpset() addsimps [asm_full_simplify
+ (simpset() addsimps [real_mult_2 RS sym] delsimps [lemma_tan_add1])
+ (read_instantiate [("x","x"),("y","x")] tan_add),numeral_2_eq_2]
+ delsimps [lemma_tan_add1]));
+qed "tan_double";
+
+Goalw [tan_def,real_divide_def] "[| 0 < x; x < pi/2 |] ==> 0 < tan x";
+by (auto_tac (claset() addSIs [sin_gt_zero2,cos_gt_zero_pi]
+ addSIs [real_mult_order,
+ real_inverse_gt_0],simpset()));
+by (rtac (CLAIM "-x < y ==> -y < (x::real)") 1 THEN Auto_tac);
+qed "tan_gt_zero";
+
+Goal "[| - pi/2 < x; x < 0 |] ==> tan x < 0";
+by (rtac (CLAIM "(0::real) < - x ==> x < 0") 1);
+by (rtac (tan_minus RS subst) 1);
+by (rtac tan_gt_zero 1);
+by (rtac (CLAIM "-x < y ==> -y < (x::real)") 2 THEN Auto_tac);
+qed "tan_less_zero";
+
+Goal "cos x ~= 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse(cos x ^ 2)";
+by (rtac lemma_DERIV_subst 1);
+by DERIV_tac;
+by (auto_tac (claset(),simpset() addsimps [real_divide_def,numeral_2_eq_2]));
+qed "lemma_DERIV_tan";
+
+Goal "cos x ~= 0 ==> DERIV tan x :> inverse(cos(x) ^ 2)";
+by (auto_tac (claset() addDs [lemma_DERIV_tan],simpset()
+ addsimps [(tan_def RS meta_eq_to_obj_eq) RS sym]));
+qed "DERIV_tan";
+Addsimps [DERIV_tan];
+
+Goalw [real_divide_def]
+ "(%x. cos(x)/sin(x)) -- pi/2 --> 0";
+by (res_inst_tac [("z1","1")] ((real_mult_0) RS subst) 1);
+by (rtac LIM_mult2 1);
+by (rtac ((real_inverse_1) RS subst) 2);
+by (rtac LIM_inverse 2);
+by (fold_tac [real_divide_def]);
+by (auto_tac (claset() addSIs [DERIV_isCont],simpset()
+ addsimps [(isCont_def RS meta_eq_to_obj_eq)
+ RS sym, cos_pi_half RS sym, sin_pi_half RS sym]
+ delsimps [cos_pi_half,sin_pi_half]));
+by (DERIV_tac THEN Auto_tac);
+qed "LIM_cos_div_sin";
+Addsimps [LIM_cos_div_sin];
+
+Goal "0 < y ==> EX x. 0 < x & x < pi/2 & y < tan x";
+by (cut_facts_tac [LIM_cos_div_sin] 1);
+by (asm_full_simp_tac (HOL_ss addsimps [LIM_def]) 1);
+by (dres_inst_tac [("x","inverse y")] spec 1);
+by (Step_tac 1);
+by (Force_tac 1);
+by (dres_inst_tac [("d1.0","s")]
+ (pi_half_gt_zero RSN (2,real_lbound_gt_zero)) 1);
+by (Step_tac 1);
+by (res_inst_tac [("x","(pi/2) - e")] exI 1);
+by (Asm_simp_tac 1);
+by (dres_inst_tac [("x","(pi/2) - e")] spec 1);
+by (auto_tac (claset(),simpset() addsimps [abs_eqI2,tan_def]));
+by (rtac (real_inverse_less_iff RS iffD1) 1);
+by (auto_tac (claset(),simpset() addsimps [real_divide_def]));
+by (rtac (real_mult_order) 1);
+by (subgoal_tac "0 < sin e" 3);
+by (subgoal_tac "0 < cos e" 3);
+by (auto_tac (claset() addIs [cos_gt_zero,sin_gt_zero2],simpset()
+ addsimps [real_inverse_distrib,abs_mult]));
+by (dres_inst_tac [("x","cos e")] (real_inverse_gt_0) 1);
+by (dres_inst_tac [("x","inverse (cos e)")] abs_eqI2 1);
+by (auto_tac (claset() addSDs [abs_eqI2],simpset() addsimps real_mult_ac));
+qed "lemma_tan_total";
+
+
+Goal "0 <= y ==> EX x. 0 <= x & x < pi/2 & tan x = y";
+by (ftac real_le_imp_less_or_eq 1);
+by (Step_tac 1 THEN Force_tac 2);
+by (dtac lemma_tan_total 1 THEN Step_tac 1);
+by (cut_inst_tac [("f","tan"),("a","0"),("b","x"),("y","y")] IVT_objl 1);
+by (auto_tac (claset() addSIs [DERIV_tan RS DERIV_isCont],simpset()));
+by (dres_inst_tac [("y","xa")] order_le_imp_less_or_eq 1);
+by (auto_tac (claset() addDs [cos_gt_zero],simpset()));
+qed "tan_total_pos";
+
+Goal "EX x. -(pi/2) < x & x < (pi/2) & tan x = y";
+by (cut_inst_tac [("y","y")] (CLAIM "0 <= (y::real) | 0 <= -y") 1);
+by (Step_tac 1);
+by (dtac tan_total_pos 1);
+by (dtac tan_total_pos 2);
+by (Step_tac 1);
+by (res_inst_tac [("x","-x")] exI 2);
+by (auto_tac (claset() addSIs [exI],simpset()));
+by (rtac (CLAIM "-x < y ==> -y < (x::real)") 1 THEN Auto_tac);
+qed "lemma_tan_total1";
+
+Goal "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y";
+by (cut_inst_tac [("y","y")] lemma_tan_total1 1);
+by (Auto_tac);
+by (cut_inst_tac [("R1.0","xa"),("R2.0","y")] real_linear 1);
+by (Auto_tac);
+by (subgoal_tac "EX z. xa < z & z < y & DERIV tan z :> 0" 1);
+by (subgoal_tac "EX z. y < z & z < xa & DERIV tan z :> 0" 3);
+by (rtac Rolle 2);
+by (rtac Rolle 7);
+by (auto_tac (claset() addSIs [DERIV_tan,DERIV_isCont,exI],simpset()
+ addsimps [differentiable_def]));
+by (TRYALL(rtac DERIV_tan));
+by (TRYALL(dtac (DERIV_tan RSN (2,DERIV_unique))));
+by (TRYALL(rtac (real_not_refl2 RS not_sym)));
+by (auto_tac (claset() addSIs [cos_gt_zero_pi],simpset()));
+by (ALLGOALS(subgoal_tac "0 < cos z"));
+by (Force_tac 1 THEN Force_tac 2);
+by (ALLGOALS(thin_tac "cos z = 0"));
+by (auto_tac (claset() addSIs [cos_gt_zero_pi],simpset()));
+qed "tan_total";
+
+Goal "[| - 1 <= y; y <= 1 |] \
+\ ==> -(pi/2) <= arcsin y & arcsin y <= pi & sin(arcsin y) = y";
+by (dtac sin_total 1);
+by (etac ex1E 2);
+by (rewtac arcsin_def);
+by (rtac someI2 2);
+by (EVERY1[assume_tac, Blast_tac, Step_tac]);
+by (rtac real_le_trans 1 THEN assume_tac 1);
+by (Force_tac 1);
+qed "arcsin_pi";
+
+Goal "[| - 1 <= y; y <= 1 |] \
+\ ==> -(pi/2) <= arcsin y & \
+\ arcsin y <= pi/2 & sin(arcsin y) = y";
+by (dtac sin_total 1 THEN assume_tac 1);
+by (etac ex1E 1);
+by (rewtac arcsin_def);
+by (rtac someI2 1);
+by (ALLGOALS(Blast_tac));
+qed "arcsin";
+
+Goal "[| - 1 <= y; y <= 1 |] ==> sin(arcsin y) = y";
+by (blast_tac (claset() addDs [arcsin]) 1);
+qed "sin_arcsin";
+Addsimps [sin_arcsin];
+
+Goal "[| -1 <= y; y <= 1 |] ==> sin(arcsin y) = y";
+by (auto_tac (claset() addIs [sin_arcsin],simpset()));
+qed "sin_arcsin2";
+Addsimps [sin_arcsin2];
+
+Goal "[| - 1 <= y; y <= 1 |] \
+\ ==> -(pi/2) <= arcsin y & arcsin y <= pi/2";
+by (blast_tac (claset() addDs [arcsin]) 1);
+qed "arcsin_bounded";
+
+Goal "[| - 1 <= y; y <= 1 |] ==> -(pi/2) <= arcsin y";
+by (blast_tac (claset() addDs [arcsin]) 1);
+qed "arcsin_lbound";
+
+Goal "[| - 1 <= y; y <= 1 |] ==> arcsin y <= pi/2";
+by (blast_tac (claset() addDs [arcsin]) 1);
+qed "arcsin_ubound";
+
+Goal "[| - 1 < y; y < 1 |] \
+\ ==> -(pi/2) < arcsin y & arcsin y < pi/2";
+by (ftac order_less_imp_le 1);
+by (forw_inst_tac [("y","y")] order_less_imp_le 1);
+by (ftac arcsin_bounded 1);
+by (Step_tac 1 THEN Asm_full_simp_tac 1);
+by (dres_inst_tac [("y","arcsin y")] order_le_imp_less_or_eq 1);
+by (dres_inst_tac [("y","pi/2")] order_le_imp_less_or_eq 2);
+by (Step_tac 1);
+by (ALLGOALS(dres_inst_tac [("f","sin")] arg_cong));
+by (Auto_tac);
+qed "arcsin_lt_bounded";
+
+Goalw [arcsin_def]
+ "[|-(pi/2) <= x; x <= pi/2 |] ==> arcsin(sin x) = x";
+by (rtac some1_equality 1);
+by (rtac sin_total 1);
+by Auto_tac;
+qed "arcsin_sin";
+
+Goal "[| - 1 <= y; y <= 1 |] \
+\ ==> 0 <= arcos y & arcos y <= pi & cos(arcos y) = y";
+by (dtac cos_total 1 THEN assume_tac 1);
+by (etac ex1E 1);
+by (rewtac arcos_def);
+by (rtac someI2 1);
+by (ALLGOALS(Blast_tac));
+qed "arcos";
+
+Goal "[| - 1 <= y; y <= 1 |] ==> cos(arcos y) = y";
+by (blast_tac (claset() addDs [arcos]) 1);
+qed "cos_arcos";
+Addsimps [cos_arcos];
+
+Goal "[| -1 <= y; y <= 1 |] ==> cos(arcos y) = y";
+by (auto_tac (claset() addIs [cos_arcos],simpset()));
+qed "cos_arcos2";
+Addsimps [cos_arcos2];
+
+Goal "[| - 1 <= y; y <= 1 |] ==> 0 <= arcos y & arcos y <= pi";
+by (blast_tac (claset() addDs [arcos]) 1);
+qed "arcos_bounded";
+
+Goal "[| - 1 <= y; y <= 1 |] ==> 0 <= arcos y";
+by (blast_tac (claset() addDs [arcos]) 1);
+qed "arcos_lbound";
+
+Goal "[| - 1 <= y; y <= 1 |] ==> arcos y <= pi";
+by (blast_tac (claset() addDs [arcos]) 1);
+qed "arcos_ubound";
+
+Goal "[| - 1 < y; y < 1 |] \
+\ ==> 0 < arcos y & arcos y < pi";
+by (ftac order_less_imp_le 1);
+by (forw_inst_tac [("y","y")] order_less_imp_le 1);
+by (ftac arcos_bounded 1);
+by (Auto_tac);
+by (dres_inst_tac [("y","arcos y")] order_le_imp_less_or_eq 1);
+by (dres_inst_tac [("y","pi")] order_le_imp_less_or_eq 2);
+by (Auto_tac);
+by (ALLGOALS(dres_inst_tac [("f","cos")] arg_cong));
+by (Auto_tac);
+qed "arcos_lt_bounded";
+
+Goalw [arcos_def] "[|0 <= x; x <= pi |] ==> arcos(cos x) = x";
+by (auto_tac (claset() addSIs [some1_equality,cos_total],simpset()));
+qed "arcos_cos";
+
+Goalw [arcos_def] "[|x <= 0; -pi <= x |] ==> arcos(cos x) = -x";
+by (auto_tac (claset() addSIs [some1_equality,cos_total],simpset()));
+qed "arcos_cos2";
+
+Goal "- (pi/2) < arctan y & \
+\ arctan y < pi/2 & tan (arctan y) = y";
+by (cut_inst_tac [("y","y")] tan_total 1);
+by (etac ex1E 1);
+by (rewtac arctan_def);
+by (rtac someI2 1);
+by (ALLGOALS(Blast_tac));
+qed "arctan";
+Addsimps [arctan];
+
+Goal "tan(arctan y) = y";
+by (Auto_tac);
+qed "tan_arctan";
+
+Goal "- (pi/2) < arctan y & arctan y < pi/2";
+by (Auto_tac);
+qed "arctan_bounded";
+
+Goal "- (pi/2) < arctan y";
+by (Auto_tac);
+qed "arctan_lbound";
+
+Goal "arctan y < pi/2";
+by (Auto_tac);
+qed "arctan_ubound";
+
+Goalw [arctan_def]
+ "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x";
+by (rtac some1_equality 1);
+by (rtac tan_total 1);
+by Auto_tac;
+qed "arctan_tan";
+
+Goal "arctan 0 = 0";
+by (rtac (asm_full_simplify (simpset())
+ (read_instantiate [("x","0")] arctan_tan)) 1);
+qed "arctan_zero_zero";
+Addsimps [arctan_zero_zero];
+
+(* ------------------------------------------------------------------------- *)
+(* Differentiation of arctan. *)
+(* ------------------------------------------------------------------------- *)
+
+Goal "cos(arctan x) ~= 0";
+by (auto_tac (claset(),simpset() addsimps [cos_zero_iff]));
+by (case_tac "n" 1);
+by (case_tac "n" 3);
+by (cut_inst_tac [("y","x")] arctan_ubound 2);
+by (cut_inst_tac [("y","x")] arctan_lbound 4);
+by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc,
+ real_add_mult_distrib,real_le_def,
+ real_mult_less_0_iff] delsimps [arctan]));
+qed "cos_arctan_not_zero";
+Addsimps [cos_arctan_not_zero];
+
+Goal "cos x ~= 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2";
+by (rtac (realpow_inverse RS subst) 1);
+by (res_inst_tac [("c1","cos(x) ^ 2")] (real_mult_right_cancel RS iffD1) 1);
+by (auto_tac (claset() addDs [realpow_not_zero], simpset() addsimps
+ [realpow_mult,real_add_mult_distrib,realpow_divide,
+ tan_def,real_mult_assoc,realpow_inverse RS sym]
+ delsimps [realpow_Suc]));
+qed "tan_sec";
+
+
+(*--------------------------------------------------------------------------*)
+(* Some more theorems- developed while at ICASE (07/2001) *)
+(*--------------------------------------------------------------------------*)
+
+Goal "sin (xa + 1 / 2 * real (Suc m) * pi) = \
+\ cos (xa + 1 / 2 * real (m) * pi)";
+by (simp_tac (HOL_ss addsimps [cos_add,sin_add,
+ real_of_nat_Suc,real_add_mult_distrib,real_add_mult_distrib2]) 1);
+by Auto_tac;
+qed "lemma_sin_cos_eq";
+Addsimps [lemma_sin_cos_eq];
+
+Goal "sin (xa + real (Suc m) * pi / 2) = \
+\ cos (xa + real (m) * pi / 2)";
+by (simp_tac (HOL_ss addsimps [cos_add,sin_add,real_divide_def,
+ real_of_nat_Suc,real_add_mult_distrib,real_add_mult_distrib2]) 1);
+by Auto_tac;
+qed "lemma_sin_cos_eq2";
+Addsimps [lemma_sin_cos_eq2];
+
+Goal "DERIV (%x. sin (x + k)) xa :> cos (xa + k)";
+by (rtac lemma_DERIV_subst 1);
+by (res_inst_tac [("f","sin"),("g","%x. x + k")] DERIV_chain2 1);
+by DERIV_tac;
+by (Simp_tac 1);
+qed "DERIV_sin_add";
+Addsimps [DERIV_sin_add];
+
+(* which further simplifies to (- 1 ^ m) !! *)
+Goal "sin ((real m + 1/2) * pi) = cos (real m * pi)";
+by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2,
+ sin_add,real_add_mult_distrib] @ real_mult_ac));
+qed "sin_cos_npi";
+Addsimps [sin_cos_npi];
+
+Goal "sin (real (Suc (2 * n)) * pi / 2) = (- 1) ^ n";
+by (cut_inst_tac [("m","n")] sin_cos_npi 1);
+by (auto_tac (claset(),HOL_ss addsimps [real_of_nat_Suc,
+ real_add_mult_distrib,real_divide_def]));
+by Auto_tac;
+qed "sin_cos_npi2";
+Addsimps [ sin_cos_npi2];
+
+Goal "cos (2 * real (n::nat) * pi) = 1";
+by (auto_tac (claset(),simpset() addsimps [cos_double,
+ real_mult_assoc,realpow_add RS sym,numeral_2_eq_2]));
+ (*FIXME: just needs x^n for literals!*)
+qed "cos_2npi";
+Addsimps [cos_2npi];
+
+Goal "cos (3 / 2 * pi) = 0";
+by (rtac (CLAIM "(1::real) + 1/2 = 3/2" RS subst) 1);
+by (rtac (real_add_mult_distrib RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [cos_add] @ real_mult_ac));
+qed "cos_3over2_pi";
+Addsimps [cos_3over2_pi];
+
+Goal "sin (2 * real (n::nat) * pi) = 0";
+by (auto_tac (claset(),simpset() addsimps [real_mult_assoc]));
+qed "sin_2npi";
+Addsimps [sin_2npi];
+
+Goal "sin (3 / 2 * pi) = - 1";
+by (rtac (CLAIM "(1::real) + 1/2 = 3/2" RS subst) 1);
+by (rtac (real_add_mult_distrib RS ssubst) 1);
+by (auto_tac (claset(),simpset() addsimps [sin_add] @real_mult_ac));
+qed "sin_3over2_pi";
+Addsimps [sin_3over2_pi];
+
+Goal "cos(xa + 1 / 2 * real (Suc m) * pi) = \
+\ -sin (xa + 1 / 2 * real (m) * pi)";
+by (simp_tac (HOL_ss addsimps [cos_add,sin_add,
+ real_of_nat_Suc,real_add_mult_distrib2,real_add_mult_distrib,
+ real_minus_mult_eq2]) 1);
+by Auto_tac;
+qed "lemma_cos_sin_eq";
+Addsimps [lemma_cos_sin_eq];
+
+Goal "cos (xa + real (Suc m) * pi / 2) = \
+\ -sin (xa + real (m) * pi / 2)";
+by (simp_tac (HOL_ss addsimps [cos_add,sin_add,real_divide_def,
+ real_of_nat_Suc,real_add_mult_distrib,real_add_mult_distrib2]) 1);
+by Auto_tac;
+qed "lemma_cos_sin_eq2";
+Addsimps [lemma_cos_sin_eq2];
+
+Goal "cos (pi * real (Suc (2 * m)) / 2) = 0";
+by (simp_tac (HOL_ss addsimps [cos_add,sin_add,real_divide_def,
+ real_of_nat_Suc,real_add_mult_distrib,real_add_mult_distrib2]) 1);
+by Auto_tac;
+qed "cos_pi_eq_zero";
+Addsimps [cos_pi_eq_zero];
+
+Goal "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)";
+by (rtac lemma_DERIV_subst 1);
+by (res_inst_tac [("f","cos"),("g","%x. x + k")] DERIV_chain2 1);
+by DERIV_tac;
+by (Simp_tac 1);
+qed "DERIV_cos_add";
+Addsimps [DERIV_cos_add];
+
+Goal "isCont cos x";
+by (rtac (DERIV_cos RS DERIV_isCont) 1);
+qed "isCont_cos";
+
+Goal "isCont sin x";
+by (rtac (DERIV_sin RS DERIV_isCont) 1);
+qed "isCont_sin";
+
+Goal "isCont exp x";
+by (rtac (DERIV_exp RS DERIV_isCont) 1);
+qed "isCont_exp";
+
+val isCont_simp = [isCont_exp,isCont_sin,isCont_cos];
+Addsimps isCont_simp;
+
+(** more theorems: e.g. used in complex geometry **)
+
+Goal "sin x = 0 ==> abs(cos x) = 1";
+by (auto_tac (claset(),simpset() addsimps [sin_zero_iff,even_mult_two_ex]));
+qed "sin_zero_abs_cos_one";
+
+Goal "(exp x = 1) = (x = 0)";
+by Auto_tac;
+by (dres_inst_tac [("f","ln")] arg_cong 1);
+by Auto_tac;
+qed "exp_eq_one_iff";
+Addsimps [exp_eq_one_iff];
+
+Goal "cos x = 1 ==> sin x = 0";
+by (cut_inst_tac [("x","x")] sin_cos_squared_add3 1);
+by Auto_tac;
+qed "cos_one_sin_zero";
+
+(*-------------------------------------------------------------------------------*)
+(* Add continuity checker in backup of theory? *)
+(*-------------------------------------------------------------------------------*)
+