# HG changeset patch
# User paulson
# Date 1090856092 -7200
# Node ID 89840837108e20b5552986677d1d29dd8a5caa75
# Parent  4b3d280ef06a85c1ff8881cd11fa3146363d62d8
converting Hyperreal/Transcendental to Isar script

diff -r 4b3d280ef06a -r 89840837108e src/HOL/Hyperreal/HTranscendental.thy
--- a/src/HOL/Hyperreal/HTranscendental.thy	Mon Jul 26 15:48:50 2004 +0200
+++ b/src/HOL/Hyperreal/HTranscendental.thy	Mon Jul 26 17:34:52 2004 +0200
@@ -60,7 +60,7 @@
 
 lemma hypreal_sqrt_gt_zero_pow2: "0 < x ==> ( *f* sqrt) (x) ^ 2 = x"
 apply (cases x)
-apply (auto intro: FreeUltrafilterNat_subset real_sqrt_gt_zero_pow2
+apply (auto intro: FreeUltrafilterNat_subset 
             simp add: hypreal_less starfun hrealpow hypreal_zero_num 
             simp del: hpowr_Suc realpow_Suc)
 done
diff -r 4b3d280ef06a -r 89840837108e src/HOL/Hyperreal/Lim.thy
--- a/src/HOL/Hyperreal/Lim.thy	Mon Jul 26 15:48:50 2004 +0200
+++ b/src/HOL/Hyperreal/Lim.thy	Mon Jul 26 17:34:52 2004 +0200
@@ -35,11 +35,11 @@
 
   (* differentiation: D is derivative of function f at x *)
   deriv:: "[real=>real,real,real] => bool"
-			    ("(DERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
+			    ("(DERIV (_)/ (_)/ :> (_))" [60, 60, 60] 60)
   "DERIV f x :> D == ((%h. (f(x + h) + -f x)/h) -- 0 --> D)"
 
   nsderiv :: "[real=>real,real,real] => bool"
-			    ("(NSDERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
+			    ("(NSDERIV (_)/ (_)/ :> (_))" [60, 60, 60] 60)
   "NSDERIV f x :> D == (\<forall>h \<in> Infinitesimal - {0}.
 			(( *f* f)(hypreal_of_real x + h) +
 			 - hypreal_of_real (f x))/h @= hypreal_of_real D)"
diff -r 4b3d280ef06a -r 89840837108e src/HOL/Hyperreal/MacLaurin_lemmas.ML
--- a/src/HOL/Hyperreal/MacLaurin_lemmas.ML	Mon Jul 26 15:48:50 2004 +0200
+++ b/src/HOL/Hyperreal/MacLaurin_lemmas.ML	Mon Jul 26 17:34:52 2004 +0200
@@ -4,6 +4,35 @@
     Description : MacLaurin series
 *)
 
+val DERIV_intros = thms"DERIV_intros";
+
+val lemma_DERIV_subst = thm"lemma_DERIV_subst";
+
+fun ARITH_PROVE str = prove_goal thy str
+                      (fn prems => [cut_facts_tac prems 1,arith_tac 1]);
+
+
+(* 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));
+
+
 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;
@@ -35,6 +64,29 @@
 (* Maclaurin's theorem with Lagrange form of remainder                       *)
 (*---------------------------------------------------------------------------*)
 
+
+
+(* 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));
+
+
 (* Annoying: Proof is now even longer due mostly to 
    change in behaviour of simplifier  since Isabelle99 *)
 Goal " [| 0 < h; 0 < n; diff 0 = f; \
@@ -123,7 +175,7 @@
     delsimps [fact_Suc,realpow_Suc]) 2);
 by (rtac DERIV_cmult 2);
 by (rtac lemma_DERIV_subst 2);
-by DERIV_tac;
+by (best_tac (claset() addIs [DERIV_chain2] addSIs DERIV_intros) 2);
 by (stac fact_Suc 2);
 by (stac real_of_nat_mult 2);
 by (simp_tac (simpset() addsimps [inverse_mult_distrib] @
diff -r 4b3d280ef06a -r 89840837108e src/HOL/Hyperreal/Transcendental.ML
--- a/src/HOL/Hyperreal/Transcendental.ML	Mon Jul 26 15:48:50 2004 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,3166 +0,0 @@
-(*  Title       : Transcendental.ML
-    Author      : Jacques D. Fleuriot
-    Copyright   : 1998,1999  University of Cambridge
-                  1999 University of Edinburgh
-    Description : Power Series
-*)
-
-fun ARITH_PROVE str = prove_goal thy str
-                      (fn prems => [cut_facts_tac prems 1,arith_tac 1]);
-
-fun multr_by_tac x i = 
-       let val cancel_thm = 
-           CLAIM "[| (0::real)<z; x*z<y*z |] ==> x<y" 
-       in
-           res_inst_tac [("z",x)] cancel_thm i 
-       end;
-
-Goalw [root_def] "root (Suc n) 0 = 0";
-by (safe_tac (claset() addSIs [some_equality,power_0_Suc] 
-    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")] zero_less_power 2);
-by (auto_tac (claset(),simpset() addsimps [zero_less_mult_iff]));
-by (res_inst_tac [("x","u"),("y","x")] linorder_cases 1);
-by (dres_inst_tac [("n1","n"),("x","u")] (zero_less_Suc RSN  (3, realpow_less)) 1);
-by (dres_inst_tac [("n1","n"),("x","x")] (zero_less_Suc RSN (3, realpow_less)) 4);
-by (auto_tac (claset(),simpset() addsimps [order_less_irrefl])); 
-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 [zero_less_power],simpset() addsimps [zero_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 [("x","u"),("y","1")] linorder_cases 1);
-by (dres_inst_tac [("n","n")] realpow_Suc_less_one 1);
-by (dres_inst_tac [("n","n")] power_gt1_lemma 4);
-by (auto_tac (claset(),simpset() addsimps [order_less_irrefl]));
-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 [nat_numeral_0_eq_0, nat_numeral_1_eq_1]
-	                addsimps [nat_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); 
-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 [("x","x"),("y","0")] linorder_less_linear 1);
-by Auto_tac;
-by (ftac (real_mult_order) 2);
-by (asm_full_simp_tac (simpset() addsimps [zero_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 [order_less_asym],
-    simpset() addsimps mult_ac@[power_mult_distrib RS sym,realpow_two_disj,
-    zero_less_power, 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 [order_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 [order_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 [zero_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,
-    zero_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 (stac real_pow_sqrt_eq_sqrt_pow 1);
-by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2, abs_mult]));
-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 (ftac real_sqrt_pow2_gt_zero 1);
-by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2, order_less_irrefl]));
-qed "real_sqrt_not_eq_zero";
-
-Goal "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x";
-by (cut_inst_tac [("n1","2"),("a1","sqrt x")] (power_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_inst_tac [("'a","real")] (zero_less_one RS 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 Safe_tac);
-by (res_inst_tac [("N","n"),("c","r")] ratio_test 1);
-by (auto_tac (claset(),
-     simpset() addsimps [abs_mult,mult_assoc RS sym] delsimps [fact_Suc]));
-by (rtac mult_right_mono 1);
-by (res_inst_tac [("b1","abs x")] (mult_commute RS ssubst) 1);
-by (stac fact_Suc 1);
-by (stac real_of_nat_mult 1);
-by (auto_tac (claset(),simpset() addsimps [abs_mult,inverse_mult_distrib]));
-by (auto_tac (claset(), simpset() addsimps 
-     [mult_assoc RS sym, abs_eqI2, positive_imp_inverse_positive]));
-by (rtac order_less_imp_le 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,
-    mult_assoc,abs_inverse]));
-by (etac order_less_trans 1);
-by (auto_tac (claset(),simpset() addsimps [mult_less_cancel_left]@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 [power_abs RS sym,
-    abs_mult,zero_le_mult_iff]));
-by (auto_tac (claset() addIs [mult_right_mono],
-    simpset() addsimps [positive_imp_inverse_positive,abs_eqI2]));
-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 [power_abs RS sym,abs_mult,
-    zero_le_mult_iff]));
-by (auto_tac (claset() addSIs [mult_right_mono],
-    simpset() addsimps [positive_imp_inverse_positive,abs_eqI2]));
-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);
-qed "lemma_STAR_sin";
-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);
-qed "lemma_STAR_cos";
-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);
-qed "lemma_STAR_cos1";
-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);
-qed "lemma_STAR_cos2";
-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 (stac lemma_realpow_diff 1);
-by (auto_tac (claset(),simpset() addsimps 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 (Asm_full_simp_tac 1); 
-by (auto_tac (claset(),simpset() delsimps [sumr_Suc]));
-by (stac sumr_Suc 1);
-by (dtac sym 1);
-by (auto_tac (claset(),simpset() addsimps [lemma_realpow_diff_sumr,
-    right_distrib,real_diff_def] @ 
-    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,
-    power_add RS sym] delsimps [sumr_Suc]));
-by (res_inst_tac [("c1","x - y")] (real_mult_left_cancel RS iffD1) 1);
-by (rtac (minus_minus RS subst) 2);
-by (stac minus_mult_left 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|.     *)
-(* ------------------------------------------------------------------------ *)
-
-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" [mult_le_cancel_left]) 1);
-by (auto_tac (claset(),
-    simpset() addsimps [mult_assoc,power_abs]));
-by (dres_inst_tac [("x","0")] spec 2 THEN Force_tac 2); 
-by (auto_tac (claset(),simpset() addsimps [abs_mult,power_abs] @ mult_ac));
-by (res_inst_tac [("a2","z ^ n")] (abs_ge_zero RS real_le_imp_less_or_eq
-    RS disjE) 1 THEN dtac sym 2);
-by (auto_tac (claset() addSIs [mult_right_mono],
-    simpset() addsimps [mult_assoc RS sym, power_abs,summable_def, power_0_left]));
-by (res_inst_tac [("x","K * inverse(1 - (abs(z) * inverse(abs x)))")] exI 1);
-by (auto_tac (claset() addSIs [sums_mult],simpset() addsimps [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 [power_abs RS sym]));
-by (subgoal_tac "x ~= 0" 1);
-by (subgoal_tac "x ~= 0" 3);
-by (auto_tac (claset(),
-     simpset() delsimps [abs_inverse, abs_mult]
-      addsimps [abs_inverse RS sym, realpow_not_zero, abs_mult RS sym,
-    power_inverse, power_mult_distrib RS sym]));
-by (auto_tac (claset() addSIs [geometric_sums],
-       simpset() addsimps [power_abs, inverse_eq_divide]));
-by (res_inst_tac [("z","abs(x)")] (CLAIM_SIMP 
-    "[|(0::real)<z; x*z<y*z |] ==> x<y" [mult_less_cancel_left]) 1);
-by (auto_tac (claset(),simpset() addsimps [abs_mult RS sym,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 [power_abs 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 [right_distrib,
-    real_diff_def,power_add RS sym] 
-    @ 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 [right_diff_distrib]
-    @ 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,
-    right_diff_distrib RS sym,real_mult_assoc] 
-    delsimps [realpow_Suc,sumr_Suc]));
-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,
-    left_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] 
-    @ 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 (stac lemma_termdiff2 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_mult]));
-by (case_tac "n" 1 THEN Auto_tac);
-by (dtac less_add_one 1);
-by (auto_tac (claset(),simpset() addsimps [power_add,real_add_assoc RS sym,
-    CLAIM_SIMP "(a * b) * c = a * (c * (b::real))" mult_ac] 
-    delsimps [sumr_Suc]));
-by (auto_tac (claset() addSIs [mult_mono],simpset()delsimps [sumr_Suc])); 
-by (auto_tac (claset() addSIs [power_mono],
-            simpset() addsimps [power_abs] 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")] zero_le_power 2);
-by (auto_tac (claset(),simpset() 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 [mult_mono], simpset() addsimps [abs_mult, power_add]));
-by (auto_tac (claset() addSIs [power_mono,zero_le_power],
-              simpset() addsimps [power_abs]));
-by (ALLGOALS(arith_tac));
-qed "lemma_termdiff3";
-
-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/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" [mult_le_cancel_left]) 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 [positive_imp_inverse_positive,
-    zero_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 [("y","K * e")] order_less_trans 2);
-by (res_inst_tac [("z","inverse K")] (CLAIM_SIMP 
-    "[|(0::real) <z; z*x<z*y |] ==> x<y" [mult_less_cancel_left]) 3);
-by (asm_full_simp_tac (simpset() addsimps [mult_assoc RS sym]) 4);
-by (Force_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [mult_less_cancel_left]) 1); 
-by (auto_tac (claset(),simpset() addsimps 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 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 [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] @ add_ac @ 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 [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,right_diff_distrib 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] @ add_ac @ 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,
-     right_distrib] @ add_ac @ mult_ac) 1);
-(* 46 *)
-by (dtac 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] @ 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 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);
-by (dtac (abs_ge_zero RS order_le_less_trans) 2);
-by (asm_full_simp_tac (simpset() addsimps mult_ac) 2);
-(* 77 *)
-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"
-     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 mult_left_mono 1);
-by (rtac (add_commute RS subst) 1);
-by (simp_tac (simpset() addsimps [mult_assoc RS sym]) 1);
-by (rtac lemma_termdiff3 1);
-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 (stac fact_Suc 1);
-by (stac real_of_nat_mult 1);
-by (stac inverse_mult_distrib 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 (stac fact_Suc 1);
-by (stac real_of_nat_mult 1);
-by (stac even_nat_Suc 1);
-by (stac inverse_mult_distrib 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 (stac fact_Suc 1);
-by (stac real_of_nat_mult 1);
-by (stac even_nat_Suc 1);
-by (stac inverse_mult_distrib 1);
-by Auto_tac;
-qed "sin_fdiffs2";
-
-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 (stac fact_Suc 1);
-by (stac real_of_nat_mult 1);
-by (stac even_nat_Suc 1);
-by (stac inverse_mult_distrib 1);
-by (res_inst_tac [("a1","real (Suc n)")] (mult_commute RS ssubst) 1);
-by (res_inst_tac [("a1","inverse(real (Suc n))")] 
-     (mult_commute RS ssubst) 1);
-by (auto_tac (claset(),simpset() addsimps [real_mult_assoc,
-    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 (stac fact_Suc 1);
-by (stac real_of_nat_mult 1);
-by (stac even_nat_Suc 1);
-by (stac inverse_mult_distrib 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_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]));
-qed "lemma_exp_ext";
-
-Goalw [exp_def] "DERIV exp x :> exp(x)";
-by (stac lemma_exp_ext 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]));
-qed "lemma_sin_ext";
-
-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]));
-qed "lemma_cos_ext";
-
-Goalw [cos_def] "DERIV sin x :> cos(x)";
-by (stac lemma_sin_ext 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 (stac lemma_cos_ext 1);
-by (auto_tac (claset(),simpset() addsimps [lemma_sin_minus,
-    cos_fdiffs2 RS sym,minus_mult_left]));
-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,zero_le_power,zero_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 (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 (minus_mult_left RS subst) 1);
-by (stac minus_mult_right 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 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 HOL_ss 1);
-by (asm_full_simp_tac (simpset() delsimps [exp_add_mult_minus] 
-    addsimps mult_ac) 1);
-qed "exp_add";
-
-Goal "0 <= exp x";
-by (res_inst_tac [("t","x")] (real_sum_of_halves RS subst) 1);
-by (stac exp_add 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))" [order_le_less]]) 1);
-qed "exp_gt_zero";
-Addsimps [exp_gt_zero];
-
-Goal "0 < inverse(exp x)";
-by (auto_tac (claset() addIs [positive_imp_inverse_positive],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,
-    right_distrib,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 (linorder_not_less RS iffD1), 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];
-
-Goal "(exp(x) <= exp(y)) = (x <= y)";
-by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym]));
-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 [le_diff_eq]));
-by (dtac (CLAIM_SIMP "x <= y ==> (0::real) <= y - x" [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 [("x","1"),("y","y")] linorder_cases 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 RS sym)],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 [("a1","ln x")] (add_left_cancel RS iffD1) 1);
-by (auto_tac (claset(),simpset() addsimps [positive_imp_inverse_positive,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 [positive_imp_inverse_positive,
-    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];
-
-Goal "[| 0 < x; 0 < y|] ==> (ln x <= ln y) = (x <= y)";
-by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym]));
-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";
-
-
-(* ------------------------------------------------------------------------ *)
-(* 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] delsimps [power_0_left]));
-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 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);
-qed "lemma_DERIV_subst";
-
-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 (stac real_add_commute 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 [("a1","(cos(x) ^ 2)")] (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 [("a1","(sin(x) ^ 2)")] (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";
-
-Goal "abs(sin x) <= 1";
-by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym]));
-by (dres_inst_tac [("n","Suc 0")] power_gt1 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];
-
-Goal "abs(cos x) <= 1";
-by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym]));
-by (dres_inst_tac [("n","Suc 0")] power_gt1 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,
-    left_distrib,right_distrib] @ 
-    mult_ac @ add_ac));
-qed "lemma_DERIV_sin_cos_add";
-
-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,
-    left_distrib,right_distrib]
-    @ mult_ac @ add_ac));
-qed "lemma_DERIV_sin_cos_minus";
-
-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);
-qed "lemma_real_of_nat_six_mult";
-
-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 (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x" 1); 
-by (asm_full_simp_tac (HOL_ss addsimps [mult_less_cancel_left]) 1); 
-by (asm_full_simp_tac (simpset() addsimps []) 1); 
-by (asm_simp_tac (simpset() addsimps [numeral_2_eq_2 RS sym, real_mult_assoc RS sym]) 1);
-by (stac (CLAIM "6 = 2 * (3::real)") 1);
-by (rtac real_mult_less_mono 1); (*mult_strict_mono would be stronger*)  
-by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc] delsimps [fact_Suc]));
-by (stac fact_Suc 1);
-by (stac fact_Suc 1);
-by (stac fact_Suc 1);
-by (stac fact_Suc 1);
-by (stac real_of_nat_mult 1);
-by (stac real_of_nat_mult 1);
-by (stac real_of_nat_mult 1);
-by (stac real_of_nat_mult 1);
-by (simp_tac (simpset() addsimps [real_divide_def,
-    inverse_mult_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 [mult_assoc,mult_less_cancel_left] 
-    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 (subgoal_tac "0 < x ^ (4 * m)" 1);
-by (asm_simp_tac (simpset() addsimps [zero_less_power]) 2); 
-by (asm_simp_tac (simpset() addsimps [mult_less_cancel_left]) 1); 
-by (rtac real_mult_less_mono 1); (*mult_strict_mono would be stronger*)
-by (ALLGOALS(Asm_simp_tac));
-by (TRYALL(rtac order_less_trans));
-by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc] delsimps [fact_Suc]));
-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 (cut_inst_tac [("x","x")] sin_gt_zero1 1);
-by (auto_tac (claset(), simpset() addsimps [cos_squared_eq, cos_double]));
-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 [zero_less_power];
-
-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 [("y",
-    "sumr 0 (Suc (Suc (Suc 0))) (%n. -((- 1) ^ n /(real (fact(2 * n))) \
-\               * 2 ^ (2 * n)))")] order_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 (stac fact_Suc 1);
-by (stac real_of_nat_mult 1);
-by (stac real_of_nat_mult 1);
-by (rtac real_mult_less_mono 1); (*mult_strict_mono would be stronger*)
-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 [("x","xa"),("y","y")] linorder_less_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 (Simp_tac 1);
-by (cut_facts_tac [pi_half_gt_zero] 1);
-by (full_simp_tac (HOL_ss addsimps [mult_zero_left, real_divide_def]) 1);
-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,left_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,left_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 (assume_tac 1); 
-by (rtac order_less_trans 1 THEN assume_tac 1);
-by (rtac pi_half_less_two 1); 
-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 "pi < 4";
-by (cut_facts_tac [pi_half_less_two] 1);
-by Auto_tac; 
-qed "pi_less_4";
-
-Goal "[| 0 < x; x < pi/2 |] ==> 0 < cos x";
-by (cut_facts_tac [pi_less_4] 1);
-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 (force_tac (claset(), simpset() addsimps []) 1); 
-by (force_tac (claset(), simpset() addsimps []) 1); 
-by (force_tac (claset(), simpset() addsimps []) 1); 
-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 [("x","x"),("y","0")] linorder_cases 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 [order_le_less,
-    cos_gt_zero_pi]));
-by Auto_tac;
-qed "cos_ge_zero";
-
-Goal "[| 0 < x; x < pi  |] ==> 0 < sin x";
-by (stac sin_cos_eq 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 [order_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 [("x","xa"),("y","y")] linorder_less_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::nat. ~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,
-    left_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,
-    left_distrib]) 1);
-by Auto_tac;
-qed "cos_zero_lemma";
-
-Goal "[| 0 <= x; sin x = 0 |] ==> \
-\     EX n::nat. 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_Suc_mult_two_ex,real_of_nat_Suc,
-    left_distrib,real_mult_assoc RS sym]));
-qed "sin_zero_lemma";
-
-(* also spoilt by numeral arithmetic *)
-Goal "(cos x = 0) = \
-\     ((EX n::nat. ~even n & (x = real n * (pi/2))) |   \
-\      (EX n::nat. ~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_Suc_mult_two_ex,real_of_nat_Suc,left_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::nat. even n & (x = real n * (pi/2))) |   \
-\      (EX n::nat. 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 [minus_mult_left]) 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() delsimps [inverse_mult_distrib]
-	     addsimps [inverse_mult_distrib RS sym] @ mult_ac));
-by (res_inst_tac [("c1","cos x * cos y")] (real_mult_right_cancel RS subst) 1);
-by (auto_tac (claset(), 
-    simpset() delsimps [inverse_mult_distrib]
-	      addsimps [mult_assoc, left_diff_distrib,cos_add]));
-qed "lemma_tan_add1";
-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 [mult_assoc, left_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 [thm"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, positive_imp_inverse_positive],simpset()));
-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 [("a1","1")] ((mult_zero_left) RS subst) 1);
-by (rtac LIM_mult2 1); 
-by (rtac (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 (inverse_less_iff_less 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 [inverse_mult_distrib,abs_mult]));
-by (dres_inst_tac [("a","cos e")] (positive_imp_inverse_positive) 1);
-by (dres_inst_tac [("x","inverse (cos e)")] abs_eqI2 1);
-by (auto_tac (claset() addSDs [abs_eqI2],simpset() addsimps 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()));
-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 [("x","xa"),("y","y")] linorder_less_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 (asm_full_simp_tac (HOL_ss addsimps [arctan]) 1); 
-qed "arctan_bounded";
-
-Goal "- (pi/2) < arctan y";
-by (Auto_tac);
-qed "arctan_lbound";
-
-Goal "arctan y < pi/2";
-by (asm_full_simp_tac (HOL_ss addsimps [arctan]) 1); 
-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, left_distrib,linorder_not_less RS sym, 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 (power_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
-    [power_mult_distrib,left_distrib,realpow_divide,
-     tan_def,real_mult_assoc,power_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,left_distrib,right_distrib]) 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,left_distrib,right_distrib]) 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 [right_distrib,
-    sin_add,left_distrib] @ 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,
-    left_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,power_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 (stac left_distrib 1);
-by (auto_tac (claset(),simpset() addsimps [cos_add] @ 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 (stac left_distrib 1);
-by (auto_tac (claset(),simpset() addsimps [sin_add] @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,right_distrib,left_distrib,
-    minus_mult_right]) 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,left_distrib,right_distrib]) 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,left_distrib,right_distrib]) 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";
-
-(*-------------------------------------------------------------------------------*)
-(* A few extra theorems                                                          *)
-(*-------------------------------------------------------------------------------*)
-
-Goal "[| 0 <= x; x < y |] ==> root(Suc n) x < root(Suc n) y";
-by (ftac order_le_less_trans 1);
-by (assume_tac 1);
-by (forw_inst_tac [("n1","n")] (real_root_pow_pos2 RS ssubst) 1);
-by (rotate_tac 1 1);
-by (assume_tac 1);
-by (forw_inst_tac [("n1","n")] (real_root_pow_pos RS ssubst) 1);
-by (rotate_tac 3 1 THEN assume_tac 1);
-by (dres_inst_tac [("y","root (Suc n) y ^ Suc n")] order_less_imp_le 1 );
-by (forw_inst_tac [("n","n")] real_root_pos_pos_le 1);
-by (forw_inst_tac [("n","n")] real_root_pos_pos 1);
-by (dres_inst_tac [("x","root (Suc n) x"),
-    ("y","root (Suc n) y")] realpow_increasing 1);
-by (assume_tac 1 THEN assume_tac 1);
-by (dres_inst_tac [("x","root (Suc n) x")] order_le_imp_less_or_eq 1);
-by Auto_tac;
-by (dres_inst_tac [("f","%x. x ^ (Suc n)")] arg_cong 1);
-by (auto_tac (claset(),simpset() addsimps [real_root_pow_pos2]
-    delsimps [realpow_Suc]));
-qed "real_root_less_mono";
-
-Goal "[| 0 <= x; x <= y |] ==> root(Suc n) x <= root(Suc n) y";
-by (dres_inst_tac [("y","y")] order_le_imp_less_or_eq 1 );
-by (auto_tac (claset() addDs [real_root_less_mono]
-    addIs [order_less_imp_le],simpset()));
-qed "real_root_le_mono";
-
-Goal "[| 0 <= x; 0 <= y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)";
-by (auto_tac (claset() addIs [real_root_less_mono],simpset()));
-by (rtac ccontr 1 THEN dtac (linorder_not_less RS iffD1) 1);
-by (dres_inst_tac [("x","y"),("n","n")] real_root_le_mono 1);
-by Auto_tac;
-qed "real_root_less_iff";
-Addsimps [real_root_less_iff];
-
-Goal "[| 0 <= x; 0 <= y |] ==> (root(Suc n) x <= root(Suc n) y) = (x <= y)";
-by (auto_tac (claset() addIs [real_root_le_mono],simpset()));
-by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
-by Auto_tac;
-by (dres_inst_tac [("x","y"),("n","n")] real_root_less_mono 1);
-by Auto_tac;
-qed "real_root_le_iff";
-Addsimps [real_root_le_iff];
-
-Goal "[| 0 <= x; 0 <= y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)";
-by (auto_tac (claset() addSIs [order_antisym],simpset()));
-by (res_inst_tac [("n1","n")] (real_root_le_iff RS iffD1) 1);
-by (res_inst_tac [("n1","n")] (real_root_le_iff RS iffD1) 4);
-by Auto_tac;
-qed "real_root_eq_iff";
-Addsimps [real_root_eq_iff];
-
-Goal "[| 0 <= x; 0 <= y; y ^ (Suc n) = x |] ==> root (Suc n) x = y";
-by (auto_tac (claset() addDs [real_root_pos2],
-    simpset() delsimps [realpow_Suc]));
-qed "real_root_pos_unique";
-
-Goal "[| 0 <= x; 0 <= y |]\
-\     ==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y";
-by (rtac real_root_pos_unique 1);
-by (auto_tac (claset() addSIs [real_root_pos_pos_le],simpset() 
-    addsimps [power_mult_distrib,zero_le_mult_iff,
-    real_root_pow_pos2] delsimps [realpow_Suc]));
-qed "real_root_mult";
-
-Goal "0 <= x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))";
-by (rtac real_root_pos_unique 1);
-by (auto_tac (claset() addIs [real_root_pos_pos_le],simpset() 
-    addsimps [power_inverse RS sym,real_root_pow_pos2] 
-    delsimps [realpow_Suc]));
-qed "real_root_inverse";
-
-Goalw [real_divide_def] 
-     "[| 0 <= x; 0 <= y |] \
-\     ==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)";
-by (auto_tac (claset(),simpset() addsimps [real_root_mult,
-    real_root_inverse]));
-qed "real_root_divide";
-
-Goalw [sqrt_def] "[| 0 <= x; x < y |] ==> sqrt(x) < sqrt(y)";
-by (auto_tac (claset() addIs [real_root_less_mono],simpset()));
-qed "real_sqrt_less_mono";
-
-Goalw [sqrt_def] "[| 0 <= x; x <= y |] ==> sqrt(x) <= sqrt(y)";
-by (auto_tac (claset() addIs [real_root_le_mono],simpset()));
-qed "real_sqrt_le_mono";
-
-Goalw [sqrt_def] "[| 0 <= x; 0 <= y |] ==> (sqrt(x) < sqrt(y)) = (x < y)";
-by Auto_tac;
-qed "real_sqrt_less_iff";
-Addsimps [real_sqrt_less_iff];
-
-Goalw [sqrt_def] "[| 0 <= x; 0 <= y |] ==> (sqrt(x) <= sqrt(y)) = (x <= y)";
-by Auto_tac;
-qed "real_sqrt_le_iff";
-Addsimps [real_sqrt_le_iff];
-
-Goalw [sqrt_def] "[| 0 <= x; 0 <= y |] ==> (sqrt(x) = sqrt(y)) = (x = y)";
-by Auto_tac;
-qed "real_sqrt_eq_iff";
-Addsimps [real_sqrt_eq_iff];
-
-Goal "(sqrt(x ^ 2 + y ^ 2) < 1) = (x ^ 2 + y ^ 2 < 1)";
-by (rtac (real_sqrt_one RS subst) 1);
-by (Step_tac 1);
-by (rtac real_sqrt_less_mono 2);
-by (dtac (rotate_prems 2 (real_sqrt_less_iff RS iffD1)) 1);
-by Auto_tac;
-qed "real_sqrt_sos_less_one_iff";
-Addsimps [real_sqrt_sos_less_one_iff];
-
-Goal "(sqrt(x ^ 2 + y ^ 2) = 1) = (x ^ 2 + y ^ 2 = 1)";
-by (rtac (real_sqrt_one RS subst) 1);
-by (Step_tac 1);
-by (dtac (rotate_prems 2 (real_sqrt_eq_iff RS iffD1)) 1);
-by Auto_tac;
-qed "real_sqrt_sos_eq_one_iff";
-Addsimps [real_sqrt_sos_eq_one_iff];
-
-Goalw [real_divide_def] "(((r::real) * a) / (r * r)) = a / r";
-by (case_tac "r=0" 1);
-by (auto_tac (claset(),simpset() addsimps [inverse_mult_distrib] @ mult_ac));
-qed "real_divide_square_eq";
-Addsimps [real_divide_square_eq];
-
-(*-------------------------------------------------------------------------------*)
-(* More theorems about sqrt, transcendental functions etc. needed in Complex.ML  *)
-(*-------------------------------------------------------------------------------*)
-
-
-Goalw [real_divide_def]
-    "0 < x ==> 0 <= x/(sqrt (x * x + y * y))";
-by (ftac ((real_sqrt_sum_squares_ge1 RSN (2,order_less_le_trans)) 
-    RS (CLAIM "0 < x ==> 0 < inverse (x::real)")) 1);
-by (rtac (real_mult_order RS order_less_imp_le) 1);
-by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2]));
-qed "lemma_real_divide_sqrt";
-
-Goal "0 < x ==> -(1::real) <= x/(sqrt (x * x + y * y))";
-by (rtac real_le_trans 1);
-by (rtac lemma_real_divide_sqrt 2);
-by Auto_tac;
-qed "lemma_real_divide_sqrt_ge_minus_one";
-
-Goal "x < 0 ==> 0 < sqrt (x * x + y * y)";
-by (rtac real_sqrt_gt_zero 1);
-by (rtac (ARITH_PROVE "[| 0 < x; 0 <= y |] ==> (0::real) < x + y") 1);
-by (auto_tac (claset(),simpset() addsimps [zero_less_mult_iff]));
-qed "real_sqrt_sum_squares_gt_zero1";
-
-Goal "0 < x ==> 0 < sqrt (x * x + y * y)";
-by (rtac real_sqrt_gt_zero 1);
-by (rtac (ARITH_PROVE "[| 0 < x; 0 <= y |] ==> (0::real) < x + y") 1);
-by (auto_tac (claset(),simpset() addsimps [zero_less_mult_iff]));
-qed "real_sqrt_sum_squares_gt_zero2";
-
-Goal "x ~= 0 ==> 0 < sqrt(x ^ 2 + y ^ 2)";
-by (cut_inst_tac [("x","x"),("y","0")] linorder_less_linear 1);
-by (auto_tac (claset() addIs [real_sqrt_sum_squares_gt_zero2,
-    real_sqrt_sum_squares_gt_zero1],simpset() addsimps [numeral_2_eq_2]));
-qed "real_sqrt_sum_squares_gt_zero3";
-
-Goal "y ~= 0 ==> 0 < sqrt(x ^ 2 + y ^ 2)";
-by (dres_inst_tac [("y","x")] real_sqrt_sum_squares_gt_zero3 1);
-by (auto_tac (claset(),simpset() addsimps [real_add_commute]));
-qed "real_sqrt_sum_squares_gt_zero3a";
-
-Goal "sqrt(x ^ 2 + y ^ 2) = x ==> y = 0";
-by (rtac ccontr 1);
-by (forw_inst_tac [("x","x")] real_sum_squares_not_zero2 1);
-by (dres_inst_tac [("f","%x. x ^ 2")] arg_cong 1);
-by (forw_inst_tac [("x","x"),("y","y")] real_sum_square_gt_zero2 1);
-by (dtac real_sqrt_gt_zero_pow2 1);
-by Auto_tac;
-qed "real_sqrt_sum_squares_eq_cancel";
-Addsimps [real_sqrt_sum_squares_eq_cancel];
-
-Goal "sqrt(x ^ 2 + y ^ 2) = y ==> x = 0";
-by (res_inst_tac [("x","y")] real_sqrt_sum_squares_eq_cancel 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1);
-qed "real_sqrt_sum_squares_eq_cancel2";
-Addsimps [real_sqrt_sum_squares_eq_cancel2];
-
-Goal "x < 0 ==> x/(sqrt (x * x + y * y)) <= 1";
-by (dtac (ARITH_PROVE "x < 0 ==> (0::real) < -x") 1);
-by (dres_inst_tac [("y","y")] 
-    lemma_real_divide_sqrt_ge_minus_one 1);
-by (dtac (ARITH_PROVE "x <= y ==> -y <= -(x::real)") 1);
-by Auto_tac;
-qed "lemma_real_divide_sqrt_le_one";
-
-Goal "x < 0 ==> -(1::real) <= x/(sqrt (x * x + y * y))";
-by (case_tac "y = 0" 1);
-by Auto_tac;
-by (ftac abs_minus_eqI2 1);
-by (auto_tac (claset(),simpset() addsimps [inverse_minus_eq]));
-by (rtac order_less_imp_le 1);
-by (res_inst_tac [("z1","sqrt(x * x + y * y)")] 
-     (real_mult_less_iff1 RS iffD1) 1);
-by (forw_inst_tac [("y2","y")] 
-    (real_sqrt_sum_squares_gt_zero1 RS real_not_refl2 
-    RS not_sym) 2);
-by (auto_tac (claset() addIs [real_sqrt_sum_squares_gt_zero1],
-    simpset() addsimps mult_ac));
-by (rtac (ARITH_PROVE "-x < y ==> -y < (x::real)") 1);
-by (cut_inst_tac [("x","-x"),("y","y")] real_sqrt_sum_squares_ge1 1);
-by (dtac real_le_imp_less_or_eq 1);
-by (Step_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [numeral_2_eq_2]) 1);
-by (dtac (sym RS real_sqrt_sum_squares_eq_cancel) 1);
-by Auto_tac;
-qed "lemma_real_divide_sqrt_ge_minus_one2";
-
-Goal "0 < x ==> x/(sqrt (x * x + y * y)) <= 1";
-by (dtac (ARITH_PROVE "0 < x ==> -x < (0::real)") 1);
-by (dres_inst_tac [("y","y")] 
-    lemma_real_divide_sqrt_ge_minus_one2 1);
-by (dtac (ARITH_PROVE "x <= y ==> -y <= -(x::real)") 1);
-by Auto_tac;
-qed "lemma_real_divide_sqrt_le_one2";
-(* was qed "lemma_real_mult_self_rinv_sqrt_squared5" *)
-
-Goal "-(1::real)<= x / sqrt (x * x + y * y)";
-by (cut_inst_tac [("x","x"),("y","0")] linorder_less_linear 1);
-by (Step_tac 1);
-by (rtac lemma_real_divide_sqrt_ge_minus_one2 1);
-by (rtac lemma_real_divide_sqrt_ge_minus_one 3);
-by Auto_tac;
-qed "cos_x_y_ge_minus_one";
-Addsimps [cos_x_y_ge_minus_one];
-
-Goal "-(1::real)<= y / sqrt (x * x + y * y)";
-by (cut_inst_tac [("x","y"),("y","x")] cos_x_y_ge_minus_one 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1);
-qed "cos_x_y_ge_minus_one1a";
-Addsimps [cos_x_y_ge_minus_one1a,
-          simplify (simpset()) cos_x_y_ge_minus_one1a];
-
-Goal "x / sqrt (x * x + y * y) <= 1";
-by (cut_inst_tac [("x","x"),("y","0")] linorder_less_linear 1);
-by (Step_tac 1);
-by (rtac lemma_real_divide_sqrt_le_one 1);
-by (rtac lemma_real_divide_sqrt_le_one2 3);
-by Auto_tac;
-qed "cos_x_y_le_one";
-Addsimps [cos_x_y_le_one];
-
-Goal "y / sqrt (x * x + y * y) <= 1";
-by (cut_inst_tac [("x","y"),("y","x")] cos_x_y_le_one 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1);
-qed "cos_x_y_le_one2";
-Addsimps [cos_x_y_le_one2];
-
-Addsimps [[cos_x_y_ge_minus_one,cos_x_y_le_one] MRS cos_arcos];
-Addsimps [[cos_x_y_ge_minus_one,cos_x_y_le_one] MRS arcos_bounded];
-
-Addsimps [[cos_x_y_ge_minus_one1a,cos_x_y_le_one2] MRS cos_arcos];
-Addsimps [[cos_x_y_ge_minus_one1a,cos_x_y_le_one2] MRS arcos_bounded];
-
-Goal "-(1::real) <= abs(x) / sqrt (x * x + y * y)";
-by (cut_inst_tac [("x","x"),("y","0")] linorder_less_linear 1);
-by (auto_tac (claset(),simpset() addsimps [abs_minus_eqI2,abs_eqI2]));
-by (dtac lemma_real_divide_sqrt_ge_minus_one 1 THEN Force_tac 1);
-qed "cos_rabs_x_y_ge_minus_one";
-
-Addsimps [cos_rabs_x_y_ge_minus_one,
-          simplify (simpset()) cos_rabs_x_y_ge_minus_one];
-
-Goal "abs(x) / sqrt (x * x + y * y) <= 1";
-by (cut_inst_tac [("x","x"),("y","0")] linorder_less_linear 1);
-by (auto_tac (claset(),simpset() addsimps [abs_minus_eqI2,abs_eqI2]));
-by (dtac lemma_real_divide_sqrt_ge_minus_one2 1 THEN Force_tac 1);
-qed "cos_rabs_x_y_le_one";
-Addsimps [cos_rabs_x_y_le_one];
-
-Addsimps [[cos_rabs_x_y_ge_minus_one,cos_rabs_x_y_le_one] MRS cos_arcos];
-Addsimps [[cos_rabs_x_y_ge_minus_one,cos_rabs_x_y_le_one] MRS arcos_bounded];
-
-Goal "-pi < 0";
-by (Simp_tac 1);
-qed "minus_pi_less_zero";
-Addsimps [minus_pi_less_zero];
-Addsimps [minus_pi_less_zero RS order_less_imp_le];
-
-Goal "[| -(1::real) <= y; y <= 1 |] ==> -pi <= arcos y";
-by (rtac real_le_trans 1);
-by (rtac arcos_lbound 2);
-by Auto_tac;
-qed "arcos_ge_minus_pi";
-
-Addsimps [[cos_x_y_ge_minus_one,cos_x_y_le_one] MRS arcos_ge_minus_pi];
-
-(* How tedious! *)
-Goal "[| x + (y::real) ~= 0; 1 - z = x/(x + y) \
-\     |] ==> z = y/(x + y)";
-by (res_inst_tac [("c1","x + y")] (real_mult_right_cancel RS iffD1) 1);
-by (forw_inst_tac [("c1","x + y")] (real_mult_right_cancel RS iffD2) 2);
-by (assume_tac 2);
-by (rotate_tac 2 2);
-by (dtac (real_mult_assoc RS subst) 2);
-by (rotate_tac 2 2);
-by (ftac (left_inverse RS subst) 2);
-by (assume_tac 2);
-by (thin_tac "(1 - z) * (x + y) = x /(x + y) * (x + y)" 2);
-by (thin_tac "1 - z = x /(x + y)" 2);
-by (auto_tac (claset(),simpset() addsimps [mult_assoc]));
-by (auto_tac (claset(),simpset() addsimps [right_distrib,
-    left_diff_distrib]));
-qed "lemma_divide_rearrange";
-
-Goal "[| 0 < x * x + y * y; \
-\       1 - sin xa ^ 2 = (x / sqrt (x * x + y * y)) ^ 2 \
-\     |] ==> sin xa ^ 2 = (y / sqrt (x * x + y * y)) ^ 2";
-by (auto_tac (claset() addIs [lemma_divide_rearrange],simpset() 
-    addsimps [realpow_divide,real_sqrt_gt_zero_pow2,
-    power2_eq_square RS sym]));
-qed "lemma_cos_sin_eq";
-
-Goal "[| 0 < x * x + y * y; \
-\       1 - cos xa ^ 2 = (y / sqrt (x * x + y * y)) ^ 2 \
-\     |] ==> cos xa ^ 2 = (x / sqrt (x * x + y * y)) ^ 2";
-by (auto_tac (claset(),
-    simpset() addsimps [realpow_divide,
-          real_sqrt_gt_zero_pow2,power2_eq_square RS sym]));
-by (rtac (real_add_commute RS subst) 1);
-by (rtac lemma_divide_rearrange 1); 
-by (asm_full_simp_tac (simpset() addsimps []) 1);
-by (asm_full_simp_tac (simpset() addsimps [add_commute]) 1);  
-qed "lemma_sin_cos_eq";
-
-Goal "[| x ~= 0; \
-\        cos xa = x / sqrt (x * x + y * y) \
-\     |] ==>  sin xa = y / sqrt (x * x + y * y) | \
-\             sin xa = - y / sqrt (x * x + y * y)";
-by (dres_inst_tac [("f","%x. x ^ 2")] arg_cong 1);
-by (forw_inst_tac [("y","y")] real_sum_square_gt_zero 1);
-by (asm_full_simp_tac (simpset() addsimps [cos_squared_eq]) 1);
-by (subgoal_tac "sin xa ^ 2 =  (y / sqrt (x * x + y * y)) ^ 2" 1);
-by (rtac lemma_cos_sin_eq 2);
-by (Force_tac 2);
-by (Asm_full_simp_tac 2);
-by (auto_tac (claset(),simpset() addsimps [realpow_two_disj,numeral_2_eq_2] 
-    delsimps [realpow_Suc]));
-qed "sin_x_y_disj";
-
-(*FIXME: remove real_sqrt_gt_zero_pow2*)
-Goal "0 <= x ==> sqrt(x) ^ 2 = x";
-by (asm_full_simp_tac (simpset() addsimps [real_sqrt_pow_abs,abs_if]) 1); 
-qed "real_sqrt_ge_zero_pow2";
-
-Goal "y ~= 0 ==> x / sqrt (x * x + y * y) ~= -(1::real)";
-by Auto_tac;
-by (dres_inst_tac [("f","%x. x ^ 2")] arg_cong 1); 
-by (asm_full_simp_tac (simpset() addsimps [power_divide,thm"real_mult_self_sum_ge_zero",real_sqrt_ge_zero_pow2]) 1); 
-by (asm_full_simp_tac (simpset() addsimps [inst "a" "1" divide_eq_eq, power2_eq_square] addsplits [split_if_asm]) 1); 
-qed "cos_not_eq_minus_one";
-
-Goalw [arcos_def] 
-  "arcos (x / sqrt (x * x + y * y)) = pi ==> y = 0";
-by (rtac ccontr 1);
-by (rtac swap 1 THEN assume_tac 2);
-by (rtac (([cos_x_y_ge_minus_one,cos_x_y_le_one] MRS cos_total) RS 
-     ((CLAIM "EX! x. P x ==> EX x. P x") RS someI2_ex)) 1);
-by (auto_tac (claset() addDs [cos_not_eq_minus_one],simpset()));
-qed "arcos_eq_pi_cancel";
-
-Goalw [real_divide_def] "x ~= 0 ==> x / sqrt (x * x + y * y) ~= 0";
-by (forw_inst_tac [("y3","y")] (real_sqrt_sum_squares_gt_zero3 
-    RS real_not_refl2 RS not_sym RS nonzero_imp_inverse_nonzero) 1);
-by (auto_tac (claset(),simpset() addsimps [power2_eq_square]));
-qed "lemma_cos_not_eq_zero";
-
-Goal "[| x ~= 0; \
-\        sin xa = y / sqrt (x * x + y * y) \
-\     |] ==>  cos xa = x / sqrt (x * x + y * y) | \
-\             cos xa = - x / sqrt (x * x + y * y)";
-by (dres_inst_tac [("f","%x. x ^ 2")] arg_cong 1);
-by (forw_inst_tac [("y","y")] real_sum_square_gt_zero 1);
-by (asm_full_simp_tac (simpset() addsimps [sin_squared_eq] 
-    delsimps [realpow_Suc]) 1);
-by (subgoal_tac "cos xa ^ 2 =  (x / sqrt (x * x + y * y)) ^ 2" 1);
-by (rtac lemma_sin_cos_eq 2);
-by (Force_tac 2);
-by (Asm_full_simp_tac 2);
-by (auto_tac (claset(),simpset() addsimps [realpow_two_disj,
-    numeral_2_eq_2] delsimps [realpow_Suc]));
-qed "cos_x_y_disj";
-
-Goal "0 < y ==> - y / sqrt (x * x + y * y) < 0";
-by (case_tac "x = 0" 1);
-by (auto_tac (claset(),simpset() addsimps [abs_eqI2]));
-by (dres_inst_tac [("y","y")] real_sqrt_sum_squares_gt_zero3 1);
-by (auto_tac (claset(),simpset() addsimps [zero_less_mult_iff,
-    real_divide_def,power2_eq_square])); 
-qed "real_sqrt_divide_less_zero";
-
-Goal "[| x ~= 0; 0 < y |] ==> EX r a. x = r * cos a & y = r * sin a";
-by (res_inst_tac [("x","sqrt(x ^ 2 + y ^ 2)")] exI 1);
-by (res_inst_tac [("x","arcos(x / sqrt (x * x + y * y))")] exI 1);
-by Auto_tac;
-by (dres_inst_tac [("y2","y")] (real_sqrt_sum_squares_gt_zero3 
-    RS real_not_refl2 RS not_sym) 1);
-by (auto_tac (claset(),simpset() addsimps [power2_eq_square]));
-by (rewtac arcos_def);
-by (cut_inst_tac [("x1","x"),("y1","y")] ([cos_x_y_ge_minus_one,cos_x_y_le_one] 
-    MRS cos_total) 1);
-by (rtac someI2_ex 1 THEN Blast_tac 1);
-by (thin_tac 
-    "EX! xa. 0 <= xa & xa <= pi & cos xa = x / sqrt (x * x + y * y)" 1);
-by (ftac sin_x_y_disj 1 THEN Blast_tac 1);
-by (dres_inst_tac [("y2","y")] (real_sqrt_sum_squares_gt_zero3 
-    RS real_not_refl2 RS not_sym) 1);
-by (auto_tac (claset(),simpset() addsimps [power2_eq_square]));
-by (dtac sin_ge_zero 1 THEN assume_tac 1);
-by (dres_inst_tac [("x","x")] real_sqrt_divide_less_zero 1 THEN Auto_tac);
-qed "polar_ex1";
-
-Goal "x * x = -(y * y) ==> y = (0::real)";
-by (auto_tac (claset() addIs [real_sum_squares_cancel],simpset()));
-qed "real_sum_squares_cancel2a";
-
-Goal "[| x ~= 0; y < 0 |] ==> EX r a. x = r * cos a & y = r * sin a";
-by (cut_inst_tac [("x","0"),("y","x")] linorder_less_linear 1);
-by Auto_tac;
-by (res_inst_tac [("x","sqrt(x ^ 2 + y ^ 2)")] exI 1);
-by (res_inst_tac [("x","arcsin(y / sqrt (x * x + y * y))")] exI 1);
-by (auto_tac (claset() addDs [real_sum_squares_cancel2a],
-    simpset() addsimps [power2_eq_square]));
-by (rewtac arcsin_def);
-by (cut_inst_tac [("x1","x"),("y1","y")] ([cos_x_y_ge_minus_one1a,
-    cos_x_y_le_one2] MRS sin_total) 1);
-by (rtac someI2_ex 1 THEN Blast_tac 1);
-by (thin_tac "EX! xa. - (pi/2) <= xa & \
-\                xa <= pi/2 & sin xa = y / sqrt (x * x + y * y)" 1);
-by (ftac ((CLAIM "0 < x ==> (x::real) ~= 0") RS cos_x_y_disj) 1 THEN Blast_tac 1);
-by Auto_tac;
-by (dtac cos_ge_zero 1 THEN Force_tac 1);
-by (dres_inst_tac [("x","y")] real_sqrt_divide_less_zero 1);
-by (auto_tac (claset(),simpset() addsimps [real_add_commute]));
-by (dtac (ARITH_PROVE "(y::real) < 0 ==> 0 < - y") 1);
-by (dtac (CLAIM "x < (0::real) ==> x ~= 0" RS polar_ex1) 1 THEN assume_tac 1);
-by (REPEAT(etac exE 1));
-by (res_inst_tac [("x","r")] exI 1);
-by (res_inst_tac [("x","-a")] exI 1);
-by Auto_tac;
-qed "polar_ex2";
-
-Goal "EX r a. x = r * cos a & y = r * sin a";
-by (case_tac "x = 0" 1);
-by Auto_tac;
-by (res_inst_tac [("x","y")] exI 1);
-by (res_inst_tac [("x","pi/2")] exI 1 THEN Auto_tac);
-by (cut_inst_tac [("x","0"),("y","y")] linorder_less_linear 1);
-by Auto_tac;
-by (res_inst_tac [("x","x")] exI 2);
-by (res_inst_tac [("x","0")] exI 2 THEN Auto_tac);
-by (ALLGOALS(blast_tac (claset() addIs [polar_ex1,polar_ex2])));
-qed "polar_Ex";
-
-Goal "abs x <= sqrt (x ^ 2 + y ^ 2)";
-by (res_inst_tac [("n","1")] realpow_increasing 1);
-by (auto_tac (claset(),simpset() addsimps [numeral_2_eq_2 RS sym]));
-qed "real_sqrt_ge_abs1";
-
-Goal "abs y <= sqrt (x ^ 2 + y ^ 2)";
-by (rtac (real_add_commute RS subst) 1);
-by (rtac real_sqrt_ge_abs1 1);
-qed "real_sqrt_ge_abs2";
-Addsimps [real_sqrt_ge_abs1,real_sqrt_ge_abs2];
-
-Goal "0 < sqrt 2";
-by (auto_tac (claset() addIs [real_sqrt_gt_zero],simpset()));
-qed "real_sqrt_two_gt_zero";
-Addsimps [real_sqrt_two_gt_zero];
-
-Goal "0 <= sqrt 2";
-by (auto_tac (claset() addIs [real_sqrt_ge_zero],simpset()));
-qed "real_sqrt_two_ge_zero";
-Addsimps [real_sqrt_two_ge_zero];
-
-Goal "1 < sqrt 2";
-by (res_inst_tac [("y","7/5")] order_less_le_trans 1);
-by (res_inst_tac [("n","1")] realpow_increasing 2);
-by (auto_tac (claset(),simpset() addsimps [real_sqrt_gt_zero_pow2,numeral_2_eq_2 RS sym] 
-    delsimps [realpow_Suc]));
-by (simp_tac (simpset() addsimps [numeral_2_eq_2]) 1);
-qed "real_sqrt_two_gt_one";
-Addsimps [real_sqrt_two_gt_one];
-
-Goal "0 < u ==> u / sqrt 2 < u";
-by (res_inst_tac [("z1","inverse u")] (real_mult_less_iff1 RS iffD1) 1);
-by Auto_tac;
-by (res_inst_tac [("z1","sqrt 2")] (real_mult_less_iff1 RS iffD1) 1);
-by Auto_tac;
-qed "lemma_real_divide_sqrt_less";
-
-(* needed for infinitely close relation over the nonstandard complex numbers *)
-Goal "[| 0 < u; x < u/2; y < u/2; 0 <= x; 0 <= y |] ==> sqrt (x ^ 2 + y ^ 2) < u";
-by (res_inst_tac [("y","u/sqrt 2")] order_le_less_trans 1);
-by (etac lemma_real_divide_sqrt_less 2);
-by (res_inst_tac [("n","1")] realpow_increasing 1);
-by (auto_tac (claset(),
-    simpset() addsimps [real_0_le_divide_iff,realpow_divide,
-    real_sqrt_gt_zero_pow2,numeral_2_eq_2 RS sym] delsimps [realpow_Suc]));
-by (res_inst_tac [("t","u ^ 2")] (real_sum_of_halves RS subst) 1);
-by (rtac add_mono 1);
-by (auto_tac (claset(),simpset() delsimps [realpow_Suc]));
-by (ALLGOALS(rtac ((CLAIM "(2::real) ^ 2 = 4") RS subst)));
-by (ALLGOALS(rtac (power_mult_distrib RS subst)));
-by (ALLGOALS(rtac power_mono));
-by Auto_tac;
-qed "lemma_sqrt_hcomplex_capprox";
-
-Addsimps [real_sqrt_sum_squares_ge_zero RS abs_eqI1];
-
-(* A few theorems involving ln and derivatives, etc *)
-
-Goal "DERIV ln z :> l ==> DERIV (%x. exp (ln x)) z :> exp (ln z) * l";
-by (etac DERIV_fun_exp 1);
-qed "lemma_DERIV_ln";
-
-Goal "0 < z ==> ( *f* (%x. exp (ln x))) z = z";
-by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
-    hypreal_zero_def,hypreal_less]));
-qed "STAR_exp_ln";
-
-Goal "[|e : Infinitesimal; 0 < x |] ==> 0 < hypreal_of_real x + e";
-by (res_inst_tac [("c1","-e")] (add_less_cancel_right RS iffD1) 1);
-by (auto_tac (claset() addIs [Infinitesimal_less_SReal],simpset()));
-qed "hypreal_add_Infinitesimal_gt_zero";
-
-Goalw [nsderiv_def,NSLIM_def] "0 < z ==> NSDERIV (%x. exp (ln x)) z :> 1";
-by Auto_tac;
-by (rtac ccontr 1);
-by (subgoal_tac "0 < hypreal_of_real z + h" 1);
-by (dtac STAR_exp_ln 1);
-by (rtac hypreal_add_Infinitesimal_gt_zero 2);
-by (dtac (CLAIM "h ~= 0 ==> h/h = (1::hypreal)") 1);
-by (auto_tac (claset(),simpset() addsimps [exp_ln_iff RS sym] 
-    delsimps [exp_ln_iff]));
-qed "NSDERIV_exp_ln_one";
-
-Goal "0 < z ==> DERIV (%x. exp (ln x)) z :> 1";
-by (auto_tac (claset() addIs [NSDERIV_exp_ln_one],
-    simpset() addsimps [NSDERIV_DERIV_iff RS sym]));
-qed "DERIV_exp_ln_one";
-
-Goal "[| 0 < z; DERIV ln z :> l |] ==>  exp (ln z) * l = 1";
-by (rtac DERIV_unique 1);
-by (rtac lemma_DERIV_ln 1);
-by (rtac DERIV_exp_ln_one 2);
-by Auto_tac;
-qed "lemma_DERIV_ln2";
-
-Goal "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/(exp (ln z))";
-by (res_inst_tac [("c1","exp(ln z)")] (real_mult_left_cancel RS iffD1) 1);
-by (auto_tac (claset() addIs [lemma_DERIV_ln2],simpset()));
-qed "lemma_DERIV_ln3";
-
-Goal "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/z";
-by (res_inst_tac [("t","z")] (exp_ln_iff RS iffD2 RS subst) 1);
-by (auto_tac (claset() addIs [lemma_DERIV_ln3],simpset()));
-qed "lemma_DERIV_ln4";
-
-(* need to rename second isCont_inverse *)
-
-Goal "[| 0 < d; ALL z. abs(z - x) <= d --> g(f(z)) = z; \
-\        ALL z. abs(z - x) <= d --> isCont f z |] \
-\     ==> isCont g (f x)";
-by (simp_tac (simpset() addsimps [isCont_iff,LIM_def]) 1);
-by (Step_tac 1);
-by (dres_inst_tac [("d1.0","r")] real_lbound_gt_zero 1);
-by (assume_tac 1 THEN Step_tac 1);
-by (subgoal_tac "ALL z. abs(z - x) <= e --> (g(f z) = z)" 1);
-by (Force_tac 2);
-by (subgoal_tac "ALL z. abs(z - x) <= e --> isCont f z" 1);
-by (Force_tac 2);
-by (dres_inst_tac [("d","e")] isCont_inj_range 1);
-by (assume_tac 2 THEN assume_tac 1);
-by (Step_tac 1);
-by (res_inst_tac [("x","ea")] exI 1);
-by Auto_tac;
-by (rotate_tac 4 1);
-by (dres_inst_tac [("x","f(x) + xa")] spec 1);
-by Auto_tac;
-by (dtac sym 1 THEN Auto_tac);
-by (arith_tac 1);
-qed "isCont_inv_fun";
-
-(*
-Goalw [isCont_def]  
-      "[| isCont f x; f x ~= 0 |] ==> isCont (%x. inverse (f x)) x";
-by (blast_tac (claset() addIs [LIM_inverse]) 1);
-qed "isCont_inverse";
-*)
-
-
-Goal "[| 0 < d; \
-\        ALL z. abs(z - x) <= d --> g(f(z)) = z; \
-\        ALL z. abs(z - x) <= d --> isCont f z |] \
-\      ==> EX e. 0 < (e::real) & \
-\            (ALL y. 0 < abs(y - f(x::real)) & abs(y - f(x)) < e --> f(g(y)) = y)";
-by (dtac isCont_inj_range 1);
-by (assume_tac 2 THEN assume_tac 1);
-by Auto_tac;
-by (res_inst_tac [("x","e")] exI 1 THEN Auto_tac);
-by (rotate_tac 2 1);
-by (dres_inst_tac [("x","y")] spec 1 THEN Auto_tac);
-qed "isCont_inv_fun_inv";
-
-
-(* Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*)
-Goal "[| f -- c --> l; 0 < l |] \
-\        ==> EX r. 0 < r & (ALL x. x ~= c & abs (c - x) < r --> 0 < f x)";
-by (auto_tac (claset(),simpset() addsimps [LIM_def]));
-by (dres_inst_tac [("x","l/2")] spec 1);
-by (Step_tac 1);
-by (Force_tac 1);
-by (res_inst_tac [("x","s")] exI 1);
-by (Step_tac 1);
-by (rotate_tac 2 1);
-by (dres_inst_tac [("x","x")] spec 1);
-by (auto_tac (claset(),HOL_ss addsimps [abs_interval_iff]));
-by (auto_tac (claset(),simpset() addsimps [CLAIM "(l::real) + -(l/2) = l/2",
-    CLAIM "(a < f + - l) = (l + a < (f::real))"]));
-qed "LIM_fun_gt_zero";
-
-Goal "[| f -- c --> l; l < 0 |] \
-\        ==> EX r. 0 < r & (ALL x. x ~= c & abs (c - x) < r --> f x < 0)";
-by (auto_tac (claset(),simpset() addsimps [LIM_def]));
-by (dres_inst_tac [("x","-l/2")] spec 1);
-by (Step_tac 1);
-by (Force_tac 1);
-by (res_inst_tac [("x","s")] exI 1);
-by (Step_tac 1);
-by (rotate_tac 2 1);
-by (dres_inst_tac [("x","x")] spec 1);
-by (auto_tac (claset(),HOL_ss addsimps [abs_interval_iff]));
-by (auto_tac (claset(),simpset() addsimps [CLAIM "(l::real) + -(l/2) = l/2",
-    CLAIM "(f + - l < a) = ((f::real) < l + a)"]));
-qed "LIM_fun_less_zero";
-
-
-Goal "[| f -- c --> l; l ~= 0 |] \
-\        ==> EX r. 0 < r & (ALL x. x ~= c & abs (c - x) < r --> f x ~= 0)";
-by (cut_inst_tac [("x","l"),("y","0")] linorder_less_linear 1);
-by Auto_tac;
-by (dtac LIM_fun_less_zero 1);
-by (dtac LIM_fun_gt_zero 3);
-by Auto_tac;
-by (ALLGOALS(res_inst_tac [("x","r")] exI));
-by Auto_tac;
-qed "LIM_fun_not_zero";
diff -r 4b3d280ef06a -r 89840837108e src/HOL/Hyperreal/Transcendental.thy
--- a/src/HOL/Hyperreal/Transcendental.thy	Mon Jul 26 15:48:50 2004 +0200
+++ b/src/HOL/Hyperreal/Transcendental.thy	Mon Jul 26 17:34:52 2004 +0200
@@ -2,10 +2,14 @@
     Author      : Jacques D. Fleuriot
     Copyright   : 1998,1999 University of Cambridge
                   1999,2001 University of Edinburgh
-    Description : Power Series, transcendental functions etc.
+    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
 *)
 
-Transcendental = NthRoot + Fact + HSeries + EvenOdd + Lim + 
+header{*Power Series, Transcendental Functions etc.*}
+
+theory Transcendental = NthRoot + Fact + HSeries + EvenOdd + Lim:
+
+(*????FOR RING_AND_FIELD*)
 
 constdefs
     root :: "[nat,real] => real"
@@ -32,18 +36,2854 @@
     "ln x == (@u. exp u = x)"
 
     pi :: "real"
-    "pi == 2 * (@x. 0 <= (x::real) & x <= 2 & cos x = 0)"
+    "pi == 2 * (@x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
 
     tan :: "real => real"
     "tan x == (sin x)/(cos x)"
 
     arcsin :: "real => real"
-    "arcsin y == (@x. -(pi/2) <= x & x <= pi/2 & sin x = y)"
+    "arcsin y == (@x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
 
     arcos :: "real => real"
-    "arcos y == (@x. 0 <= x & x <= pi & cos x = y)"
+    "arcos y == (@x. 0 \<le> x & x \<le> pi & cos x = y)"
      
     arctan :: "real => real"
     "arctan y == (@x. -(pi/2) < x & x < pi/2 & tan x = y)"
+
+
+lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
+apply (unfold root_def)
+apply (safe intro!: some_equality power_0_Suc elim!: realpow_zero_zero)
+done
+
+lemma real_root_pow_pos: 
+     "0 < x ==> (root(Suc n) x) ^ (Suc n) = x"
+apply (unfold root_def)
+apply (drule_tac n = n in realpow_pos_nth2)
+apply (auto intro: someI2)
+done
+
+lemma real_root_pow_pos2: "0 \<le> x ==> (root(Suc n) x) ^ (Suc n) = x"
+by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos)
+
+lemma real_root_pos: 
+     "0 < x ==> root(Suc n) (x ^ (Suc n)) = x"
+apply (unfold root_def)
+apply (rule some_equality)
+apply (frule_tac [2] n = n in zero_less_power)
+apply (auto simp add: zero_less_mult_iff)
+apply (rule_tac x = u and y = x in linorder_cases)
+apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less])
+apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less])
+apply (auto simp add: order_less_irrefl)
+done
+
+lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x"
+by (auto dest!: real_le_imp_less_or_eq real_root_pos)
+
+lemma real_root_pos_pos: 
+     "0 < x ==> 0 \<le> root(Suc n) x"
+apply (unfold root_def)
+apply (drule_tac n = n in realpow_pos_nth2)
+apply (safe, rule someI2)
+apply (auto intro!: order_less_imp_le dest: zero_less_power simp add: zero_less_mult_iff)
+done
+
+lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x"
+by (auto dest!: real_le_imp_less_or_eq dest: real_root_pos_pos)
+
+lemma real_root_one [simp]: "root (Suc n) 1 = 1"
+apply (unfold root_def)
+apply (rule some_equality, auto)
+apply (rule ccontr)
+apply (rule_tac x = u and y = 1 in linorder_cases)
+apply (drule_tac n = n in realpow_Suc_less_one)
+apply (drule_tac [4] n = n in power_gt1_lemma)
+apply (auto simp add: order_less_irrefl)
+done
+
+
+subsection{*Square Root*}
+
+(*lcp: needed now because 2 is a binary numeral!*)
+lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"
+apply (simp (no_asm) del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 add: nat_numeral_0_eq_0 [symmetric])
+done
+
+lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
+by (unfold sqrt_def, auto)
+
+lemma real_sqrt_one [simp]: "sqrt 1 = 1"
+by (unfold sqrt_def, auto)
+
+lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
+apply (unfold sqrt_def)
+apply (rule iffI) 
+ apply (cut_tac r = "root 2 x" in realpow_two_le)
+ apply (simp add: numeral_2_eq_2)
+apply (subst numeral_2_eq_2)
+apply (erule real_root_pow_pos2)
+done
+
+lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>"
+by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])
+
+lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x"
+by (simp add: real_sqrt_pow2_iff)
+
+lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>"
+by (rule real_sqrt_pow2_iff [THEN iffD2], arith)
+
+lemma real_pow_sqrt_eq_sqrt_pow: 
+      "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)"
+apply (unfold sqrt_def)
+apply (subst numeral_2_eq_2)
+apply (auto intro: real_root_pow_pos2 [THEN ssubst] real_root_pos2 [THEN ssubst] simp del: realpow_Suc)
+done
+
+lemma real_pow_sqrt_eq_sqrt_abs_pow2:
+     "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
+by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])
+
+lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>"
+apply (rule real_sqrt_abs_abs [THEN subst])
+apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst])
+apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric])
+apply (assumption, arith)
+done
+
+lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
+apply auto
+apply (cut_tac x = x and y = 0 in linorder_less_linear)
+apply (simp add: zero_less_mult_iff)
+done
+
+lemma real_mult_self_eq_zero_iff [simp]: "(r * r = 0) = (r = (0::real))"
+by auto
+
+lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"
+apply (unfold sqrt_def root_def)
+apply (subst numeral_2_eq_2)
+apply (drule realpow_pos_nth2 [where n=1], safe)
+apply (rule someI2, auto)
+done
+
+lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
+by (auto intro: real_sqrt_gt_zero simp add: order_le_less)
+
+
+(*we need to prove something like this:
+lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r"
+apply (case_tac n, simp) 
+apply (unfold root_def) 
+apply (rule someI2 [of _ r], safe)
+apply (auto simp del: realpow_Suc dest: power_inject_base)
+*)
+
+lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r"
+apply (unfold sqrt_def root_def) 
+apply (rule someI2 [of _ r], auto) 
+apply (auto simp add: numeral_2_eq_2 simp del: realpow_Suc 
+            dest: power_inject_base) 
+done
+
+lemma real_sqrt_mult_distrib: 
+     "[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)"
+apply (rule sqrt_eqI)
+apply (simp add: power_mult_distrib)  
+apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
+done
+
+lemma real_sqrt_mult_distrib2: "[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) =  sqrt(x) * sqrt(y)"
+by (auto intro: real_sqrt_mult_distrib simp add: order_le_less)
+
+lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
+by (auto intro!: real_sqrt_ge_zero)
+
+lemma real_sqrt_sum_squares_mult_ge_zero [simp]: "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
+by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
+
+lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
+     "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
+by (auto simp add: real_sqrt_pow2_iff zero_le_mult_iff simp del: realpow_Suc)
+
+lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>"
+apply (rule abs_realpow_two [THEN subst])
+apply (rule real_sqrt_abs_abs [THEN subst])
+apply (subst real_pow_sqrt_eq_sqrt_pow)
+apply (auto simp add: numeral_2_eq_2 abs_mult)
+done
+
+lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
+apply (rule realpow_two [THEN subst])
+apply (subst numeral_2_eq_2 [symmetric])
+apply (rule real_sqrt_abs)
+done
+
+lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
+by simp
+
+lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
+apply (frule real_sqrt_pow2_gt_zero)
+apply (auto simp add: numeral_2_eq_2 order_less_irrefl)
+done
+
+lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
+by (cut_tac n1 = 2 and a1 = "sqrt x" in power_inverse [symmetric], auto)
+
+lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
+apply (drule real_le_imp_less_or_eq)
+apply (auto dest: real_sqrt_not_eq_zero)
+done
+
+lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x = 0))"
+by (auto simp add: real_sqrt_eq_zero_cancel)
+
+lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
+apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe)
+apply (rule real_le_trans)
+apply (auto simp del: realpow_Suc)
+apply (rule_tac n = 1 in realpow_increasing)
+apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc)
+done
+
+lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)"
+apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
+done
+
+lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
+apply (rule_tac n = 1 in realpow_increasing)
+apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp 
+            del: realpow_Suc)
+done
+
+
+subsection{*Exponential Function*}
+
+lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
+apply (cut_tac 'a = real in zero_less_one [THEN dense], safe)
+apply (cut_tac x = r in reals_Archimedean3, auto)
+apply (drule_tac x = "\<bar>x\<bar>" in spec, safe)
+apply (rule_tac N = n and c = r in ratio_test)
+apply (auto simp add: abs_mult mult_assoc [symmetric] simp del: fact_Suc)
+apply (rule mult_right_mono)
+apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult)
+apply (auto simp add: abs_mult inverse_mult_distrib)
+apply (auto simp add: mult_assoc [symmetric] abs_eqI2 positive_imp_inverse_positive)
+apply (rule order_less_imp_le)
+apply (rule_tac z1 = "real (Suc na) " in real_mult_less_iff1 [THEN iffD1])
+apply (auto simp add: real_not_refl2 [THEN not_sym] mult_assoc abs_inverse)
+apply (erule order_less_trans)
+apply (auto simp add: mult_less_cancel_left mult_ac)
+done
+
+
+lemma summable_sin: 
+     "summable (%n.  
+           (if even n then 0  
+           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
+                x ^ n)"
+apply (unfold real_divide_def)
+apply (rule_tac g = " (%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n) " in summable_comparison_test)
+apply (rule_tac [2] summable_exp)
+apply (rule_tac x = 0 in exI)
+apply (auto simp add: power_abs [symmetric] abs_mult zero_le_mult_iff)
+done
+
+lemma summable_cos: 
+      "summable (%n.  
+           (if even n then  
+           (- 1) ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
+apply (unfold real_divide_def)
+apply (rule_tac g = " (%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n) " in summable_comparison_test)
+apply (rule_tac [2] summable_exp)
+apply (rule_tac x = 0 in exI)
+apply (auto simp add: power_abs [symmetric] abs_mult zero_le_mult_iff)
+done
+
+lemma lemma_STAR_sin [simp]: "(if even n then 0  
+       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
+apply (induct_tac "n", auto)
+done
+
+lemma lemma_STAR_cos [simp]: "0 < n -->  
+      (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
+apply (induct_tac "n", auto)
+done
+
+lemma lemma_STAR_cos1 [simp]: "0 < n -->  
+      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
+apply (induct_tac "n", auto)
+done
+
+lemma lemma_STAR_cos2 [simp]: "sumr 1 n (%n. if even n  
+                    then (- 1) ^ (n div 2)/(real (fact n)) *  
+                          0 ^ n  
+                    else 0) = 0"
+apply (induct_tac "n")
+apply (case_tac [2] "n", auto)
+done
+
+lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
+apply (unfold exp_def)
+apply (rule summable_exp [THEN summable_sums])
+done
+
+lemma sin_converges: 
+      "(%n. (if even n then 0  
+            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
+                 x ^ n) sums sin(x)"
+apply (unfold sin_def)
+apply (rule summable_sin [THEN summable_sums])
+done
+
+lemma cos_converges: 
+      "(%n. (if even n then  
+           (- 1) ^ (n div 2)/(real (fact n))  
+           else 0) * x ^ n) sums cos(x)"
+apply (unfold cos_def)
+apply (rule summable_cos [THEN summable_sums])
+done
+
+lemma lemma_realpow_diff [rule_format (no_asm)]: "p \<le> n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y"
+apply (induct_tac "n", auto)
+apply (subgoal_tac "p = Suc n")
+apply (simp (no_asm_simp), auto)
+apply (drule sym)
+apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric] 
+       del: realpow_Suc)
+done
+
+
+subsection{*Properties of Power Series*}
+
+lemma lemma_realpow_diff_sumr:
+     "sumr 0 (Suc n) (%p. (x ^ p) * y ^ ((Suc n) - p)) =  
+      y * sumr 0 (Suc n) (%p. (x ^ p) * (y ^ (n - p)))"
+apply (auto simp add: sumr_mult simp del: sumr_Suc)
+apply (rule sumr_subst)
+apply (intro strip)
+apply (subst lemma_realpow_diff)
+apply (auto simp add: mult_ac)
+done
+
+lemma lemma_realpow_diff_sumr2: "x ^ (Suc n) - y ^ (Suc n) =  
+      (x - y) * sumr 0 (Suc n) (%p. (x ^ p) * (y ^(n - p)))"
+apply (induct_tac "n", simp)
+apply (auto simp del: sumr_Suc)
+apply (subst sumr_Suc)
+apply (drule sym)
+apply (auto simp add: lemma_realpow_diff_sumr right_distrib real_diff_def mult_ac simp del: sumr_Suc)
+done
+
+lemma lemma_realpow_rev_sumr: "sumr 0 (Suc n) (%p. (x ^ p) * (y ^ (n - p))) =  
+      sumr 0 (Suc n) (%p. (x ^ (n - p)) * (y ^ p))"
+apply (case_tac "x = y")
+apply (auto simp add: mult_commute power_add [symmetric] simp del: sumr_Suc)
+apply (rule_tac c1 = "x - y" in real_mult_left_cancel [THEN iffD1])
+apply (rule_tac [2] minus_minus [THEN subst], simp)
+apply (subst minus_mult_left)
+apply (simp add: lemma_realpow_diff_sumr2 [symmetric] del: sumr_Suc)
+done
+
+text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
+x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
+
+lemma powser_insidea:
+     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
+      ==> summable (%n. abs(f(n)) * (z ^ n))"
+apply (drule summable_LIMSEQ_zero)
+apply (drule convergentI)
+apply (simp add: Cauchy_convergent_iff [symmetric])
+apply (drule Cauchy_Bseq)
+apply (simp add: Bseq_def, safe)
+apply (rule_tac g = "%n. K * abs (z ^ n) * inverse (abs (x ^ n))" in summable_comparison_test)
+apply (rule_tac x = 0 in exI, safe)
+apply (subgoal_tac "0 < abs (x ^ n) ")
+apply (rule_tac c="abs (x ^ n)" in mult_right_le_imp_le) 
+apply (auto simp add: mult_assoc power_abs)
+ prefer 2
+ apply (drule_tac x = 0 in spec, force)
+apply (auto simp add: abs_mult power_abs mult_ac)
+apply (rule_tac a2 = "z ^ n" 
+       in abs_ge_zero [THEN real_le_imp_less_or_eq, THEN disjE])
+apply (auto intro!: mult_right_mono simp add: mult_assoc [symmetric] power_abs summable_def power_0_left)
+apply (rule_tac x = "K * inverse (1 - (\<bar>z\<bar> * inverse (\<bar>x\<bar>))) " in exI)
+apply (auto intro!: sums_mult simp add: mult_assoc)
+apply (subgoal_tac "abs (z ^ n) * inverse (\<bar>x\<bar> ^ n) = (\<bar>z\<bar> * inverse (\<bar>x\<bar>)) ^ n")
+apply (auto simp add: power_abs [symmetric])
+apply (subgoal_tac "x \<noteq> 0")
+apply (subgoal_tac [3] "x \<noteq> 0")
+apply (auto simp del: abs_inverse abs_mult simp add: abs_inverse [symmetric] realpow_not_zero abs_mult [symmetric] power_inverse power_mult_distrib [symmetric])
+apply (auto intro!: geometric_sums simp add: power_abs inverse_eq_divide)
+apply (rule_tac c="\<bar>x\<bar>" in mult_right_less_imp_less) 
+apply (auto simp add: abs_mult [symmetric] mult_assoc)
+done
+
+lemma powser_inside: "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
+      ==> summable (%n. f(n) * (z ^ n))"
+apply (drule_tac z = "\<bar>z\<bar>" in powser_insidea)
+apply (auto intro: summable_rabs_cancel simp add: power_abs [symmetric])
+done
+
+
+subsection{*Differentiation of Power Series*}
+
+text{*Lemma about distributing negation over it*}
+lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
+by (simp add: diffs_def)
+
+text{*Show that we can shift the terms down one*}
+lemma lemma_diffs:
+     "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))"
+apply (induct_tac "n")
+apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)
+done
+
+lemma lemma_diffs2: "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 simp add: lemma_diffs)
+
+
+lemma diffs_equiv: "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
+      (%n. real n * c(n) * (x ^ (n - Suc 0))) sums  
+         (suminf(%n. (diffs c)(n) * (x ^ n)))"
+apply (subgoal_tac " (%n. real n * c (n) * (x ^ (n - Suc 0))) ----> 0")
+apply (rule_tac [2] LIMSEQ_imp_Suc)
+apply (drule summable_sums) 
+apply (auto simp add: sums_def)
+apply (drule_tac X="(\<lambda>n. \<Sum>n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff)
+apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])
+apply (simp add: diffs_def summable_LIMSEQ_zero)
+done
+
+
+subsection{*Term-by-Term Differentiability of Power Series*}
+
+lemma lemma_termdiff1:
+     "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))))"
+apply (rule sumr_subst)
+apply (auto simp add: right_distrib real_diff_def power_add [symmetric] mult_ac)
+done
+
+lemma less_add_one: "m < n ==> (\<exists>d. n = m + d + Suc 0)"
+by (simp add: less_iff_Suc_add)
+
+lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"
+by arith
+
+
+lemma lemma_termdiff2: " h \<noteq> 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))))"
+apply (rule real_mult_left_cancel [THEN iffD1], simp (no_asm_simp))
+apply (simp add: right_diff_distrib mult_ac)
+apply (simp add: mult_assoc [symmetric])
+apply (case_tac "n")
+apply (auto simp add: lemma_realpow_diff_sumr2 
+                      right_diff_distrib [symmetric] mult_assoc
+            simp del: realpow_Suc sumr_Suc)
+apply (auto simp add: lemma_realpow_rev_sumr simp del: sumr_Suc)
+apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 
+                sumdiff lemma_termdiff1 sumr_mult)
+apply (auto intro!: sumr_subst simp add: real_diff_def real_add_assoc)
+apply (simp add: diff_minus [symmetric] less_iff_Suc_add) 
+apply (auto simp add: sumr_mult lemma_realpow_diff_sumr2 mult_ac simp
+                 del: sumr_Suc realpow_Suc)
+done
+
+lemma lemma_termdiff3: "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; abs (z + h) \<le> K |]  
+      ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))  
+          \<le> real n * real (n - Suc 0) * K ^ (n - 2) * \<bar>h\<bar>"
+apply (subst lemma_termdiff2, assumption)
+apply (simp add: abs_mult mult_commute) 
+apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 
+apply (rule sumr_rabs [THEN real_le_trans])
+apply (simp add: mult_assoc [symmetric])
+apply (simp add: mult_commute [of _ "real (n - Suc 0)"])
+apply (auto intro!: sumr_bound2 simp add: abs_mult)
+apply (case_tac "n", auto)
+apply (drule less_add_one)
+(*CLAIM_SIMP " (a * b * c = a * (c * (b::real))" mult_ac]*)
+apply clarify 
+apply (subgoal_tac "K ^ p * K ^ d * real (Suc (Suc (p + d))) =
+                    K ^ p * (real (Suc (Suc (p + d))) * K ^ d)") 
+apply (simp (no_asm_simp) add: power_add del: sumr_Suc)
+apply (auto intro!: mult_mono simp del: sumr_Suc)
+apply (auto intro!: power_mono simp add: power_abs simp del: sumr_Suc)
+apply (rule_tac j = "real (Suc d) * (K ^ d) " in real_le_trans)
+apply (subgoal_tac [2] "0 \<le> K")
+apply (drule_tac [2] n = d in zero_le_power)
+apply (auto simp del: sumr_Suc)
+apply (rule sumr_rabs [THEN real_le_trans])
+apply (rule sumr_bound2, auto dest!: less_add_one intro!: mult_mono simp add: abs_mult power_add)
+apply (auto intro!: power_mono zero_le_power simp add: power_abs, arith+)
+done
+
+lemma lemma_termdiff4: 
+  "[| 0 < k;  
+      (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> abs(f h) \<le> K * \<bar>h\<bar>) |]  
+   ==> f -- 0 --> 0"
+apply (unfold LIM_def, auto)
+apply (subgoal_tac "0 \<le> K")
+apply (drule_tac [2] x = "k/2" in spec)
+apply (frule_tac [2] real_less_half_sum)
+apply (drule_tac [2] real_gt_half_sum)
+apply (auto simp add: abs_eqI2)
+apply (rule_tac [2] c = "k/2" in mult_right_le_imp_le)
+apply (auto intro: abs_ge_zero [THEN real_le_trans])
+apply (drule real_le_imp_less_or_eq, auto)
+apply (subgoal_tac "0 < (r * inverse K) * inverse 2")
+apply (rule_tac [2] real_mult_order)+
+apply (drule_tac ?d1.0 = "r * inverse K * inverse 2" and ?d2.0 = k in real_lbound_gt_zero)
+apply (auto simp add: positive_imp_inverse_positive zero_less_mult_iff)
+apply (rule_tac [2] y="\<bar>f (k / 2)\<bar> * 2" in order_trans, auto)
+apply (rule_tac x = e in exI, auto)
+apply (rule_tac y = "K * \<bar>x\<bar>" in order_le_less_trans)
+apply (rule_tac [2] y = "K * e" in order_less_trans)
+apply (rule_tac [3] c = "inverse K" in mult_right_less_imp_less, force)
+apply (simp add: mult_less_cancel_left)
+apply (auto simp add: mult_ac)
+done
+
+lemma lemma_termdiff5: "[| 0 < k;  
+            summable f;  
+            \<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k -->  
+                    (\<forall>n. abs(g(h) (n::nat)) \<le> (f(n) * \<bar>h\<bar>)) |]  
+         ==> (%h. suminf(g h)) -- 0 --> 0"
+apply (drule summable_sums)
+apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> abs (suminf (g h)) \<le> suminf f * \<bar>h\<bar>")
+apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
+apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
+ prefer 2
+ apply (simp add: summable_def) 
+ apply (rule_tac x = "suminf f * \<bar>h\<bar>" in exI)
+ apply (drule_tac c = "\<bar>h\<bar>" in sums_mult)
+ apply (simp add: mult_ac) 
+apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
+ apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
+  apply (rule_tac [2] x = 0 in exI, auto)
+apply (rule_tac j = "suminf (%n. abs (g h n))" in real_le_trans)
+apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult])
+done
+
+
+
+text{* FIXME: Long proofs*}
+
+lemma termdiffs_aux:
+     "[|summable (\<lambda>n. diffs (diffs c) n * K ^ n); \<bar>x\<bar> < \<bar>K\<bar> |]
+    ==> (\<lambda>h. suminf
+             (\<lambda>n. c n *
+                  (((x + h) ^ n - x ^ n) * inverse h -
+                   real n * x ^ (n - Suc 0))))
+       -- 0 --> 0"
+apply (drule dense, safe)
+apply (frule real_less_sum_gt_zero)
+apply (drule_tac
+         f = "%n. abs (c n) * real n * real (n - Suc 0) * (r ^ (n - 2))" 
+     and g = "%h n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) 
+                             - (real n * (x ^ (n - Suc 0))))" 
+       in lemma_termdiff5)
+apply (auto simp add: add_commute)
+apply (subgoal_tac "summable (%n. \<bar>diffs (diffs c) n\<bar> * (r ^ n))")
+apply (rule_tac [2] x = K in powser_insidea, auto)
+apply (subgoal_tac [2] "\<bar>r\<bar> = r", auto)
+apply (rule_tac [2] y1 = "\<bar>x\<bar>" in order_trans [THEN abs_eqI1], auto)
+apply (simp add: diffs_def mult_assoc [symmetric])
+apply (subgoal_tac
+        "\<forall>n. real (Suc n) * real (Suc (Suc n)) * \<bar>c (Suc (Suc n))\<bar> * (r ^ n) 
+              = diffs (diffs (%n. \<bar>c n\<bar>)) n * (r ^ n) ") 
+apply auto
+apply (drule diffs_equiv)
+apply (drule sums_summable)
+apply (simp_all add: diffs_def) 
+apply (simp add: diffs_def mult_ac)
+apply (subgoal_tac " (%n. real n * (real (Suc n) * (abs (c (Suc n)) * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * abs (c m) * inverse r) n * (r ^ n))")
+apply auto
+  prefer 2
+  apply (rule ext)
+  apply (simp add: diffs_def) 
+  apply (case_tac "n", auto)
+txt{*23*}
+   apply (drule abs_ge_zero [THEN order_le_less_trans])
+   apply (simp add: mult_ac) 
+  apply (drule abs_ge_zero [THEN order_le_less_trans])
+  apply (simp add: mult_ac) 
+ apply (drule diffs_equiv)
+ apply (drule sums_summable)
+apply (subgoal_tac
+          "summable
+            (\<lambda>n. real n * (real (n - Suc 0) * \<bar>c n\<bar> * inverse r) *
+                 r ^ (n - Suc 0)) =
+           summable
+            (\<lambda>n. real n * (\<bar>c n\<bar> * (real (n - Suc 0) * r ^ (n - 2))))")
+apply simp 
+apply (rule_tac f = summable in arg_cong, rule ext)
+txt{*33*}
+apply (case_tac "n", auto)
+apply (case_tac "nat", auto)
+apply (drule abs_ge_zero [THEN order_le_less_trans], auto) 
+apply (drule abs_ge_zero [THEN order_le_less_trans])
+apply (simp add: mult_assoc)
+apply (rule mult_left_mono)
+apply (rule add_commute [THEN subst])
+apply (simp (no_asm) add: mult_assoc [symmetric])
+apply (rule lemma_termdiff3)
+apply (auto intro: abs_triangle_ineq [THEN order_trans], arith)
+done
+
+
+lemma termdiffs: 
+    "[| summable(%n. c(n) * (K ^ n));  
+        summable(%n. (diffs c)(n) * (K ^ n));  
+        summable(%n. (diffs(diffs c))(n) * (K ^ n));  
+        \<bar>x\<bar> < \<bar>K\<bar> |]  
+     ==> DERIV (%x. suminf (%n. c(n) * (x ^ n)))  x :>  
+             suminf (%n. (diffs c)(n) * (x ^ n))"
+apply (unfold deriv_def)
+apply (rule_tac g = "%h. suminf (%n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h) " in LIM_trans)
+apply (simp add: LIM_def, safe)
+apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
+apply (auto simp add: less_diff_eq)
+apply (drule abs_triangle_ineq [THEN order_le_less_trans])
+apply (rule_tac y = 0 in order_le_less_trans, auto)
+apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%n. (c n) * ( (x + xa) ^ n))) ")
+apply (auto intro!: summable_sums)
+apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
+apply (auto simp add: add_commute)
+apply (drule_tac x="(\<lambda>n. c n * (xa + x) ^ n)" in sums_diff, assumption) 
+apply (drule_tac x = " (%n. c n * (xa + x) ^ n - c n * x ^ n) " and c = "inverse xa" in sums_mult)
+apply (rule sums_unique [symmetric])
+apply (simp add: diff_def real_divide_def add_ac mult_ac)
+apply (rule LIM_zero_cancel)
+apply (rule_tac g = "%h. suminf (%n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) - (real n * (x ^ (n - Suc 0))))) " in LIM_trans)
+ prefer 2 apply (blast intro: termdiffs_aux) 
+apply (simp add: LIM_def, safe)
+apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
+apply (auto simp add: less_diff_eq)
+apply (drule abs_triangle_ineq [THEN order_le_less_trans])
+apply (rule_tac y = 0 in order_le_less_trans, auto)
+apply (subgoal_tac "summable (%n. (diffs c) (n) * (x ^ n))")
+apply (rule_tac [2] powser_inside, auto)
+apply (drule_tac c = c and x = x in diffs_equiv)
+apply (frule sums_unique, auto)
+apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%n. (c n) * ( (x + xa) ^ n))) ")
+apply safe
+apply (auto intro!: summable_sums)
+apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
+apply (auto simp add: add_commute)
+apply (frule_tac x = " (%n. c n * (xa + x) ^ n) " and y = " (%n. c n * x ^ n) " in sums_diff, assumption)
+apply (simp add: suminf_diff [OF sums_summable sums_summable] 
+               right_diff_distrib [symmetric])
+apply (frule_tac x = " (%n. c n * ((xa + x) ^ n - x ^ n))" and c = "inverse xa" in sums_mult)
+apply (simp add: sums_summable [THEN suminf_mult2])
+apply (frule_tac x = " (%n. inverse xa * (c n * ((xa + x) ^ n - x ^ n))) " and y = " (%n. real n * c n * x ^ (n - Suc 0))" in sums_diff)
+apply assumption
+apply (subst minus_equation_iff, simp (no_asm))  
+apply (simp add: suminf_diff [OF sums_summable sums_summable] add_ac mult_ac)
+apply (rule_tac f = suminf in arg_cong)
+apply (rule ext)
+apply (simp add: diff_def right_distrib add_ac mult_ac)
+done
+
+
+subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 
+
+lemma exp_fdiffs: 
+      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
+apply (unfold diffs_def)
+apply (rule ext)
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult)
+apply (subst inverse_mult_distrib)
+apply (auto simp add: mult_assoc [symmetric])
+done
+
+lemma sin_fdiffs: 
+      "diffs(%n. if even n then 0  
+           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))  
+       = (%n. if even n then  
+                 (- 1) ^ (n div 2)/(real (fact n))  
+              else 0)"
+apply (unfold diffs_def real_divide_def)
+apply (rule ext)
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult)
+apply (subst even_nat_Suc)
+apply (subst inverse_mult_distrib, auto)
+done
+
+lemma sin_fdiffs2: 
+       "diffs(%n. if even n then 0  
+           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n  
+       = (if even n then  
+                 (- 1) ^ (n div 2)/(real (fact n))  
+              else 0)"
+apply (unfold diffs_def real_divide_def)
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult)
+apply (subst even_nat_Suc)
+apply (subst inverse_mult_distrib, auto)
+done
+
+lemma cos_fdiffs: 
+      "diffs(%n. if even n then  
+                 (- 1) ^ (n div 2)/(real (fact n)) else 0)  
+       = (%n. - (if even n then 0  
+           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n))))"
+apply (unfold diffs_def real_divide_def)
+apply (rule ext)
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult)
+apply (auto simp add: mult_assoc odd_Suc_mult_two_ex)
+done
+
+
+lemma cos_fdiffs2: 
+      "diffs(%n. if even n then  
+                 (- 1) ^ (n div 2)/(real (fact n)) else 0) n 
+       = - (if even n then 0  
+           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n)))"
+apply (unfold diffs_def real_divide_def)
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult) 
+apply (auto simp add: mult_assoc odd_Suc_mult_two_ex)
+done
+
+text{*Now at last we can get the derivatives of exp, sin and cos*}
+
+lemma lemma_sin_minus:
+     "- sin x =
+         suminf(%n. - ((if even n then 0 
+                  else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
+by (auto intro!: sums_unique sums_minus sin_converges)
+
+lemma lemma_exp_ext: "exp = (%x. suminf (%n. inverse (real (fact n)) * x ^ n))"
+by (auto intro!: ext simp add: exp_def)
+
+lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
+apply (unfold exp_def)
+apply (subst lemma_exp_ext)
+apply (subgoal_tac "DERIV (%u. suminf (%n. inverse (real (fact n)) * u ^ n)) x :> suminf (%n. diffs (%n. inverse (real (fact n))) n * x ^ n) ")
+apply (rule_tac [2] K = "1 + \<bar>x\<bar> " in termdiffs)
+apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs, arith)
+done
+
+lemma lemma_sin_ext:
+     "sin = (%x. suminf(%n. 
+                   (if even n then 0  
+                       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
+                   x ^ n))"
+by (auto intro!: ext simp add: sin_def)
+
+lemma lemma_cos_ext:
+     "cos = (%x. suminf(%n. 
+                   (if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
+                   x ^ n))"
+by (auto intro!: ext simp add: cos_def)
+
+lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
+apply (unfold cos_def)
+apply (subst lemma_sin_ext)
+apply (auto simp add: sin_fdiffs2 [symmetric])
+apply (rule_tac K = "1 + \<bar>x\<bar> " in termdiffs)
+apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs, arith)
+done
+
+lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
+apply (subst lemma_cos_ext)
+apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
+apply (rule_tac K = "1 + \<bar>x\<bar> " in termdiffs)
+apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus, arith)
+done
+
+
+subsection{*Properties of the Exponential Function*}
+
+lemma exp_zero [simp]: "exp 0 = 1"
+proof -
+  have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
+        suminf (\<lambda>n. inverse (real (fact n)) * 0 ^ n)"
+    by (rule series_zero [rule_format, THEN sums_unique],
+        case_tac "m", auto)
+  thus ?thesis by (simp add:  exp_def) 
+qed
+
+lemma exp_ge_add_one_self [simp]: "0 \<le> x ==> (1 + x) \<le> exp(x)"
+apply (drule real_le_imp_less_or_eq, auto)
+apply (unfold exp_def)
+apply (rule real_le_trans)
+apply (rule_tac [2] n = 2 and f = " (%n. inverse (real (fact n)) * x ^ n) " in series_pos_le)
+apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
+done
+
+lemma exp_gt_one [simp]: "0 < x ==> 1 < exp x"
+apply (rule order_less_le_trans)
+apply (rule_tac [2] exp_ge_add_one_self, auto)
+done
+
+lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
+proof -
+  have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
+    by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_Id DERIV_const) 
+  thus ?thesis by (simp add: o_def)
+qed
+
+lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
+proof -
+  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
+    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_Id) 
+  thus ?thesis by (simp add: o_def)
+qed
+
+lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
+proof -
+  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
+       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
+    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
+  thus ?thesis by simp
+qed
+
+lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
+proof -
+  have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
+  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
+    by (rule DERIV_isconst_all) 
+  thus ?thesis by simp
+qed
+
+lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
+proof -
+  have "exp (x + 0) * exp (- x) = exp 0" by (rule exp_add_mult_minus) 
+  thus ?thesis by simp
+qed
+
+lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
+by (simp add: mult_commute)
+
+
+lemma exp_minus: "exp(-x) = inverse(exp(x))"
+by (auto intro: inverse_unique [symmetric])
+
+lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
+proof -
+  have "exp x * exp y = exp x * (exp (x + y) * exp (- x))" by simp
+  thus ?thesis by (simp (no_asm_simp) add: mult_ac)
+qed
+
+text{*Proof: because every exponential can be seen as a square.*}
+lemma exp_ge_zero [simp]: "0 \<le> exp x"
+apply (rule_tac t = x in real_sum_of_halves [THEN subst])
+apply (subst exp_add, auto)
+done
+
+lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
+apply (cut_tac x = x in exp_mult_minus2)
+apply (auto simp del: exp_mult_minus2)
+done
+
+lemma exp_gt_zero [simp]: "0 < exp x"
+by (simp add: order_less_le)
+
+lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
+by (auto intro: positive_imp_inverse_positive)
+
+lemma abs_exp_cancel [simp]: "abs(exp x) = exp x"
+by (auto simp add: abs_eqI2)
+
+lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
+apply (induct_tac "n")
+apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
+done
+
+lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
+apply (unfold real_diff_def real_divide_def)
+apply (simp (no_asm) add: exp_add exp_minus)
+done
+
+
+lemma exp_less_mono:
+  assumes xy: "x < y" shows "exp x < exp y"
+proof -
+  have "1 < exp (y + - x)"
+    by (rule real_less_sum_gt_zero [THEN exp_gt_one])
+  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
+    by (auto simp add: exp_add exp_minus)
+  thus ?thesis
+    by (simp add: divide_inverse [symmetric] pos_less_divide_eq)
+qed
+
+lemma exp_less_cancel: "exp x < exp y ==> x < y"
+apply (rule ccontr) 
+apply (simp add: linorder_not_less order_le_less) 
+apply (auto dest: exp_less_mono)
+done
+
+lemma exp_less_cancel_iff [iff]: "(exp(x) < exp(y)) = (x < y)"
+by (auto intro: exp_less_mono exp_less_cancel)
+
+lemma exp_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
+by (auto simp add: linorder_not_less [symmetric])
+
+lemma exp_inj_iff [iff]: "(exp x = exp y) = (x = y)"
+by (simp add: order_eq_iff)
+
+lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x) = y"
+apply (rule IVT)
+apply (auto intro: DERIV_exp [THEN DERIV_isCont] simp add: le_diff_eq)
+apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
+apply simp 
+apply (rule exp_ge_add_one_self, simp)
+done
+
+lemma exp_total: "0 < y ==> \<exists>x. exp x = y"
+apply (rule_tac x = 1 and y = y in linorder_cases)
+apply (drule order_less_imp_le [THEN lemma_exp_total])
+apply (rule_tac [2] x = 0 in exI)
+apply (frule_tac [3] real_inverse_gt_one)
+apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
+apply (rule_tac x = "-x" in exI)
+apply (simp add: exp_minus)
+done
+
+
+subsection{*Properties of the Logarithmic Function*}
+
+lemma ln_exp[simp]: "ln(exp x) = x"
+by (simp add: ln_def)
+
+lemma exp_ln_iff[simp]: "(exp(ln x) = x) = (0 < x)"
+apply (auto dest: exp_total)
+apply (erule subst, simp) 
+done
+
+lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
+apply (rule exp_inj_iff [THEN iffD1])
+apply (frule real_mult_order)
+apply (auto simp add: exp_add exp_ln_iff [symmetric] simp del: exp_inj_iff exp_ln_iff)
+done
+
+lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
+apply (simp only: exp_ln_iff [symmetric])
+apply (erule subst)+
+apply simp 
+done
+
+lemma ln_one[simp]: "ln 1 = 0"
+by (rule exp_inj_iff [THEN iffD1], auto)
+
+lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
+apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
+apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
+done
+
+lemma ln_div: 
+    "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
+apply (unfold real_divide_def)
+apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
+done
+
+lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
+apply (simp only: exp_ln_iff [symmetric])
+apply (erule subst)+
+apply simp 
+done
+
+lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
+by (auto simp add: linorder_not_less [symmetric])
+
+lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
+by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
+
+lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
+apply (rule ln_exp [THEN subst])
+apply (rule ln_le_cancel_iff [THEN iffD2], auto)
+done
+
+lemma ln_less_self [simp]: "0 < x ==> ln x < x"
+apply (rule order_less_le_trans)
+apply (rule_tac [2] ln_add_one_self_le_self)
+apply (rule ln_less_cancel_iff [THEN iffD2], auto)
+done
+
+lemma ln_ge_zero:
+  assumes x: "1 \<le> x" shows "0 \<le> ln x"
+proof -
+  have "0 < x" using x by arith
+  hence "exp 0 \<le> exp (ln x)"
+    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
+  thus ?thesis by (simp only: exp_le_cancel_iff)
+qed
+
+lemma ln_ge_zero_imp_ge_one:
+  assumes ln: "0 \<le> ln x" 
+      and x:  "0 < x"
+  shows "1 \<le> x"
+proof -
+  from ln have "ln 1 \<le> ln x" by simp
+  thus ?thesis by (simp add: x del: ln_one) 
+qed
+
+lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
+by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
+
+lemma ln_gt_zero:
+  assumes x: "1 < x" shows "0 < ln x"
+proof -
+  have "0 < x" using x by arith
+  hence "exp 0 < exp (ln x)"
+    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
+  thus ?thesis  by (simp only: exp_less_cancel_iff)
+qed
+
+lemma ln_gt_zero_imp_gt_one:
+  assumes ln: "0 < ln x" 
+      and x:  "0 < x"
+  shows "1 < x"
+proof -
+  from ln have "ln 1 < ln x" by simp
+  thus ?thesis by (simp add: x del: ln_one) 
+qed
+
+lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
+by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
+
+lemma ln_not_eq_zero [simp]: "[| 0 < x; x \<noteq> 1 |] ==> ln x \<noteq> 0"
+apply safe
+apply (drule exp_inj_iff [THEN iffD2])
+apply (drule exp_ln_iff [THEN iffD2], auto)
+done
+
+lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
+apply (rule exp_less_cancel_iff [THEN iffD1])
+apply (auto simp add: exp_ln_iff [symmetric] simp del: exp_ln_iff)
+done
+
+lemma exp_ln_eq: "exp u = x ==> ln x = u"
+by auto
+
+
+subsection{*Basic Properties of the Trigonometric Functions*}
+
+lemma sin_zero [simp]: "sin 0 = 0"
+by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
+         simp add: sin_def sums_def simp del: power_0_left)
+
+lemma lemma_series_zero2: "(\<forall>m. n \<le> m --> f m = 0) --> f sums sumr 0 n f"
+by (auto intro: series_zero)
+
+lemma cos_zero [simp]: "cos 0 = 1"
+apply (unfold cos_def)
+apply (rule sums_unique [symmetric])
+apply (cut_tac n = 1 and f = " (%n. (if even n then (- 1) ^ (n div 2) / (real (fact n)) else 0) * 0 ^ n) " in lemma_series_zero2)
+apply auto
+done
+
+lemma DERIV_sin_sin_mult [simp]:
+     "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
+by (rule DERIV_mult, auto)
+
+lemma DERIV_sin_sin_mult2 [simp]:
+     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
+apply (cut_tac x = x in DERIV_sin_sin_mult)
+apply (auto simp add: mult_assoc)
+done
+
+lemma DERIV_sin_realpow2 [simp]:
+     "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
+by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
+
+lemma DERIV_sin_realpow2a [simp]:
+     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
+by (auto simp add: numeral_2_eq_2)
+
+lemma DERIV_cos_cos_mult [simp]:
+     "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
+by (rule DERIV_mult, auto)
+
+lemma DERIV_cos_cos_mult2 [simp]:
+     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
+apply (cut_tac x = x in DERIV_cos_cos_mult)
+apply (auto simp add: mult_ac)
+done
+
+lemma DERIV_cos_realpow2 [simp]:
+     "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
+by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
+
+lemma DERIV_cos_realpow2a [simp]:
+     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
+by (auto simp add: numeral_2_eq_2)
+
+lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
+by auto
+
+lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
+apply (rule lemma_DERIV_subst)
+apply (rule DERIV_cos_realpow2a, auto)
+done
+
+(* most useful *)
+lemma DERIV_cos_cos_mult3 [simp]: "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
+apply (rule lemma_DERIV_subst)
+apply (rule DERIV_cos_cos_mult2, auto)
+done
+
+lemma DERIV_sin_circle_all: 
+     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
+             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
+apply (unfold real_diff_def, safe)
+apply (rule DERIV_add)
+apply (auto simp add: numeral_2_eq_2)
+done
+
+lemma DERIV_sin_circle_all_zero [simp]: "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
+by (cut_tac DERIV_sin_circle_all, auto)
+
+lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
+apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
+apply (auto simp add: numeral_2_eq_2)
+done
+
+lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
+apply (subst real_add_commute)
+apply (simp (no_asm) del: realpow_Suc)
+done
+
+lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
+apply (cut_tac x = x in sin_cos_squared_add2)
+apply (auto simp add: numeral_2_eq_2)
+done
+
+lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
+apply (rule_tac a1 = "(cos x)\<twosuperior> " in add_right_cancel [THEN iffD1])
+apply (simp del: realpow_Suc)
+done
+
+lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
+apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
+apply (simp del: realpow_Suc)
+done
+
+lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
+by arith
+
+lemma abs_sin_le_one [simp]: "abs(sin x) \<le> 1"
+apply (auto simp add: linorder_not_less [symmetric])
+apply (drule_tac n = "Suc 0" in power_gt1)
+apply (auto simp del: realpow_Suc)
+apply (drule_tac r1 = "cos x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
+apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
+done
+
+lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
+apply (insert abs_sin_le_one [of x]) 
+apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
+done
+
+lemma sin_le_one [simp]: "sin x \<le> 1"
+apply (insert abs_sin_le_one [of x]) 
+apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
+done
+
+lemma abs_cos_le_one [simp]: "abs(cos x) \<le> 1"
+apply (auto simp add: linorder_not_less [symmetric])
+apply (drule_tac n = "Suc 0" in power_gt1)
+apply (auto simp del: realpow_Suc)
+apply (drule_tac r1 = "sin x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
+apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
+done
+
+lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
+apply (insert abs_cos_le_one [of x]) 
+apply (simp add: abs_le_interval_iff del: abs_cos_le_one) 
+done
+
+lemma cos_le_one [simp]: "cos x \<le> 1"
+apply (insert abs_cos_le_one [of x]) 
+apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
+done
+
+lemma DERIV_fun_pow: "DERIV g x :> m ==>  
+      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
+apply (rule lemma_DERIV_subst)
+apply (rule_tac f = " (%x. x ^ n) " in DERIV_chain2)
+apply (rule DERIV_pow, auto)
+done
+
+lemma DERIV_fun_exp: "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
+apply (rule lemma_DERIV_subst)
+apply (rule_tac f = exp in DERIV_chain2)
+apply (rule DERIV_exp, auto)
+done
+
+lemma DERIV_fun_sin: "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
+apply (rule lemma_DERIV_subst)
+apply (rule_tac f = sin in DERIV_chain2)
+apply (rule DERIV_sin, auto)
+done
+
+lemma DERIV_fun_cos: "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
+apply (rule lemma_DERIV_subst)
+apply (rule_tac f = cos in DERIV_chain2)
+apply (rule DERIV_cos, auto)
+done
+
+lemmas DERIV_intros = DERIV_Id DERIV_const DERIV_cos DERIV_cmult 
+                    DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
+                    DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
+                    DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
+                    DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos 
+                    DERIV_Id DERIV_const DERIV_cos
+
+(* lemma *)
+lemma lemma_DERIV_sin_cos_add: "\<forall>x.  
+         DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
+               (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
+apply (safe, rule lemma_DERIV_subst)
+apply (best intro!: DERIV_intros intro: DERIV_chain2) 
+  --{*replaces the old @{text DERIV_tac}*}
+apply (auto simp add: real_diff_def left_distrib right_distrib mult_ac add_ac)
+done
+
+lemma sin_cos_add [simp]:
+     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
+      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
+apply (cut_tac y = 0 and x = x and y7 = y 
+       in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
+apply (auto simp add: numeral_2_eq_2)
+done
+
+lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
+apply (cut_tac x = x and y = y in sin_cos_add)
+apply (auto dest!: real_sum_squares_cancel_a 
+            simp add: numeral_2_eq_2 simp del: sin_cos_add)
+done
+
+lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
+apply (cut_tac x = x and y = y in sin_cos_add)
+apply (auto dest!: real_sum_squares_cancel_a
+            simp add: numeral_2_eq_2 simp del: sin_cos_add)
+done
+
+lemma lemma_DERIV_sin_cos_minus: "\<forall>x.  
+          DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
+apply (safe, rule lemma_DERIV_subst)
+apply (best intro!: DERIV_intros intro: DERIV_chain2) 
+apply (auto simp add: real_diff_def left_distrib right_distrib mult_ac add_ac)
+done
+
+lemma sin_cos_minus [simp]: "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
+apply (cut_tac y = 0 and x = x in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
+apply (auto simp add: numeral_2_eq_2)
+done
+
+lemma sin_minus [simp]: "sin (-x) = -sin(x)"
+apply (cut_tac x = x in sin_cos_minus)
+apply (auto dest!: real_sum_squares_cancel_a simp add: numeral_2_eq_2 simp del: sin_cos_minus)
+done
+
+lemma cos_minus [simp]: "cos (-x) = cos(x)"
+apply (cut_tac x = x in sin_cos_minus)
+apply (auto dest!: real_sum_squares_cancel_a simp add: numeral_2_eq_2 simp del: sin_cos_minus)
+done
+
+lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
+apply (unfold real_diff_def)
+apply (simp (no_asm) add: sin_add)
+done
+
+lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
+by (simp add: sin_diff mult_commute)
+
+lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
+apply (unfold real_diff_def)
+apply (simp (no_asm) add: cos_add)
+done
+
+lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
+by (simp add: cos_diff mult_commute)
+
+lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
+by (cut_tac x = x and y = x in sin_add, auto)
+
+
+lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
+apply (cut_tac x = x and y = x in cos_add)
+apply (auto simp add: numeral_2_eq_2)
+done
+
+
+subsection{*The Constant Pi*}
+
+text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; 
+   hence define pi.*}
+
+lemma sin_paired:
+     "(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) 
+      sums  sin x"
+proof -
+  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
+            (if even k then 0
+             else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
+            x ^ k) 
+	sums
+	suminf (\<lambda>n. (if even n then 0
+		     else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
+	            x ^ n)" 
+    by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) 
+  thus ?thesis by (simp add: mult_ac sin_def)
+qed
+
+lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
+apply (subgoal_tac 
+       "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
+              (- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) 
+     sums suminf (\<lambda>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
+ prefer 2
+ apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) 
+apply (rotate_tac 2)
+apply (drule sin_paired [THEN sums_unique, THEN ssubst])
+apply (auto simp del: fact_Suc realpow_Suc)
+apply (frule sums_unique)
+apply (auto simp del: fact_Suc realpow_Suc)
+apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
+apply (auto simp del: fact_Suc realpow_Suc)
+apply (erule sums_summable)
+apply (case_tac "m=0")
+apply (simp (no_asm_simp))
+apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")
+apply (simp only: mult_less_cancel_left, simp)
+apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
+apply (subgoal_tac "x*x < 2*3", simp) 
+apply (rule mult_strict_mono)
+apply (auto simp add: real_of_nat_Suc simp del: fact_Suc)
+apply (subst fact_Suc)
+apply (subst fact_Suc)
+apply (subst fact_Suc)
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult)
+apply (subst real_of_nat_mult)
+apply (subst real_of_nat_mult)
+apply (subst real_of_nat_mult)
+apply (simp (no_asm) add: real_divide_def inverse_mult_distrib del: fact_Suc)
+apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
+apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) 
+apply (auto simp add: mult_assoc simp del: fact_Suc)
+apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) 
+apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
+apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") 
+apply (erule ssubst)+
+apply (auto simp del: fact_Suc)
+apply (subgoal_tac "0 < x ^ (4 * m) ")
+ prefer 2 apply (simp only: zero_less_power) 
+apply (simp (no_asm_simp) add: mult_less_cancel_left)
+apply (rule mult_strict_mono)
+apply (simp_all (no_asm_simp))
+done
+
+lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
+by (auto intro: sin_gt_zero)
+
+lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
+apply (cut_tac x = x in sin_gt_zero1)
+apply (auto simp add: cos_squared_eq cos_double)
+done
+
+lemma cos_paired:
+     "(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
+proof -
+  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
+            (if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
+            x ^ k) 
+        sums
+	suminf
+	 (\<lambda>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
+	      x ^ n)" 
+    by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) 
+  thus ?thesis by (simp add: mult_ac cos_def)
+qed
+
+declare zero_less_power [simp]
+
+lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
+by simp
+
+lemma cos_two_less_zero: "cos (2) < 0"
+apply (cut_tac x = 2 in cos_paired)
+apply (drule sums_minus)
+apply (rule neg_less_iff_less [THEN iffD1]) 
+apply (frule sums_unique, auto)
+apply (rule_tac y = "sumr 0 (Suc (Suc (Suc 0))) (%n. - ((- 1) ^ n / (real (fact (2 * n))) * 2 ^ (2 * n))) " in order_less_trans)
+apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
+apply (simp (no_asm) add: mult_assoc del: sumr_Suc)
+apply (rule sumr_pos_lt_pair)
+apply (erule sums_summable, safe)
+apply (simp (no_asm) add: real_divide_def mult_assoc [symmetric] del: fact_Suc)
+apply (rule real_mult_inverse_cancel2)
+apply (rule real_of_nat_fact_gt_zero)+
+apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
+apply (subst fact_lemma) 
+apply (subst fact_Suc)
+apply (subst real_of_nat_mult)
+apply (erule ssubst, subst real_of_nat_mult)
+apply (rule real_mult_less_mono, force)
+prefer 2 apply force
+apply (rule_tac [2] real_of_nat_fact_gt_zero)
+apply (rule real_of_nat_less_iff [THEN iffD2])
+apply (rule fact_less_mono, auto)
+done
+declare cos_two_less_zero [simp]
+declare cos_two_less_zero [THEN real_not_refl2, simp]
+declare cos_two_less_zero [THEN order_less_imp_le, simp]
+
+lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
+apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
+apply (rule_tac [2] IVT2)
+apply (auto intro: DERIV_isCont DERIV_cos)
+apply (cut_tac x = xa and y = y in linorder_less_linear)
+apply (rule ccontr)
+apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
+apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
+apply (drule_tac f = cos in Rolle)
+apply (drule_tac [5] f = cos in Rolle)
+apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
+apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
+apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) 
+apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) 
+done
+    
+lemma pi_half: "pi/2 = (@x. 0 \<le> x & x \<le> 2 & cos x = 0)"
+by (simp add: pi_def)
+
+lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
+apply (rule cos_is_zero [THEN ex1E])
+apply (auto intro!: someI2 simp add: pi_half)
+done
+
+lemma pi_half_gt_zero: "0 < pi / 2"
+apply (rule cos_is_zero [THEN ex1E])
+apply (auto simp add: pi_half)
+apply (rule someI2, blast, safe)
+apply (drule_tac y = xa in real_le_imp_less_or_eq)
+apply (safe, simp)
+done
+declare pi_half_gt_zero [simp]
+declare pi_half_gt_zero [THEN real_not_refl2, THEN not_sym, simp]
+declare pi_half_gt_zero [THEN order_less_imp_le, simp]
+
+lemma pi_half_less_two: "pi / 2 < 2"
+apply (rule cos_is_zero [THEN ex1E])
+apply (auto simp add: pi_half)
+apply (rule someI2, blast, safe)
+apply (drule_tac x = xa in order_le_imp_less_or_eq)
+apply (safe, simp)
+done
+declare pi_half_less_two [simp]
+declare pi_half_less_two [THEN real_not_refl2, simp]
+declare pi_half_less_two [THEN order_less_imp_le, simp]
+
+lemma pi_gt_zero [simp]: "0 < pi"
+apply (rule_tac c="inverse 2" in mult_less_imp_less_right) 
+apply (cut_tac pi_half_gt_zero, simp_all)
+done
+
+lemma pi_neq_zero [simp]: "pi \<noteq> 0"
+by (rule pi_gt_zero [THEN real_not_refl2, THEN not_sym])
+
+lemma pi_not_less_zero [simp]: "~ (pi < 0)"
+apply (insert pi_gt_zero)
+apply (blast elim: order_less_asym) 
+done
+
+lemma pi_ge_zero [simp]: "0 \<le> pi"
+by (auto intro: order_less_imp_le)
+
+lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
+by auto
+
+lemma sin_pi_half [simp]: "sin(pi/2) = 1"
+apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
+apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
+apply (auto simp add: numeral_2_eq_2)
+done
+
+lemma cos_pi [simp]: "cos pi = -1"
+by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
+
+lemma sin_pi [simp]: "sin pi = 0"
+by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
+
+lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
+apply (unfold real_diff_def)
+apply (simp (no_asm) add: cos_add)
+done
+
+lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
+apply (simp (no_asm) add: cos_add)
+done
+declare minus_sin_cos_eq [symmetric, simp]
+
+lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
+apply (unfold real_diff_def)
+apply (simp (no_asm) add: sin_add)
+done
+declare sin_cos_eq [symmetric, simp] cos_sin_eq [symmetric, simp]
+
+lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
+apply (simp (no_asm) add: sin_add)
+done
+
+lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
+apply (simp (no_asm) add: sin_add)
+done
+
+lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
+apply (simp (no_asm) add: cos_add)
+done
+
+lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
+by (simp add: sin_add cos_double)
+
+lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
+by (simp add: cos_add cos_double)
+
+lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
+apply (induct_tac "n")
+apply (auto simp add: real_of_nat_Suc left_distrib)
+done
+
+lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
+apply (induct_tac "n")
+apply (auto simp add: real_of_nat_Suc left_distrib)
+done
+
+lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
+apply (cut_tac n = n in sin_npi)
+apply (auto simp add: mult_commute simp del: sin_npi)
+done
+
+lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
+by (simp add: cos_double)
+
+lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
+apply (simp (no_asm))
+done
+
+lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
+apply (rule sin_gt_zero, assumption)
+apply (rule order_less_trans, assumption)
+apply (rule pi_half_less_two)
+done
+
+lemma sin_less_zero: 
+  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
+proof -
+  have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) 
+  thus ?thesis by simp
+qed
+
+lemma pi_less_4: "pi < 4"
+by (cut_tac pi_half_less_two, auto)
+
+lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
+apply (cut_tac pi_less_4)
+apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
+apply (force intro: DERIV_isCont DERIV_cos)
+apply (cut_tac cos_is_zero, safe)
+apply (rename_tac y z)
+apply (drule_tac x = y in spec)
+apply (drule_tac x = "pi/2" in spec, simp) 
+done
+
+lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
+apply (rule_tac x = x and y = 0 in linorder_cases)
+apply (rule cos_minus [THEN subst])
+apply (rule cos_gt_zero)
+apply (auto intro: cos_gt_zero)
+done
+ 
+lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
+apply (auto simp add: order_le_less cos_gt_zero_pi)
+apply (subgoal_tac "x = pi/2", auto) 
+done
+
+lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
+apply (subst sin_cos_eq)
+apply (rotate_tac 1)
+apply (drule real_sum_of_halves [THEN ssubst])
+apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
+done
+
+lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
+by (auto simp add: order_le_less sin_gt_zero_pi)
+
+lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
+apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
+apply (rule_tac [2] IVT2)
+apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
+apply (cut_tac x = xa and y = y in linorder_less_linear)
+apply (rule ccontr, auto)
+apply (drule_tac f = cos in Rolle)
+apply (drule_tac [5] f = cos in Rolle)
+apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
+            dest!: DERIV_cos [THEN DERIV_unique] 
+            simp add: differentiable_def)
+apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
+done
+
+lemma sin_total:
+     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
+apply (rule ccontr)
+apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")
+apply (erule swap)
+apply (simp del: minus_sin_cos_eq [symmetric])
+apply (cut_tac y="-y" in cos_total, simp) apply simp 
+apply (erule ex1E)
+apply (rule_tac a = "x - (pi/2) " in ex1I)
+apply (simp (no_asm) add: real_add_assoc)
+apply (rotate_tac 3)
+apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) 
+done
+
+lemma reals_Archimedean4:
+     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
+apply (auto dest!: reals_Archimedean3)
+apply (drule_tac x = x in spec, clarify) 
+apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
+ prefer 2 apply (erule LeastI) 
+apply (case_tac "LEAST m::nat. x < real m * y", simp) 
+apply (subgoal_tac "~ x < real nat * y")
+ prefer 2 apply (rule not_less_Least, simp, force)  
+done
+
+(* Pre Isabelle99-2 proof was simpler- numerals arithmetic 
+   now causes some unwanted re-arrangements of literals!   *)
+lemma cos_zero_lemma: "[| 0 \<le> x; cos x = 0 |] ==>  
+      \<exists>n::nat. ~even n & x = real n * (pi/2)"
+apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
+apply (subgoal_tac "0 \<le> x - real n * pi & (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
+apply safe
+  prefer 3 apply (simp add: cos_diff) 
+ prefer 2 apply (simp add: real_of_nat_Suc left_distrib) 
+apply (simp add: cos_diff)
+apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
+apply (rule_tac [2] cos_total, safe)
+apply (drule_tac x = "x - real n * pi" in spec)
+apply (drule_tac x = "pi/2" in spec)
+apply (simp add: cos_diff)
+apply (rule_tac x = "Suc (2 * n) " in exI)
+apply (simp add: real_of_nat_Suc left_distrib, auto)
+done
+
+lemma sin_zero_lemma: "[| 0 \<le> x; sin x = 0 |] ==>  
+      \<exists>n::nat. even n & x = real n * (pi/2)"
+apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
+ apply (clarify, rule_tac x = "n - 1" in exI)
+ apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
+apply (rule cos_zero_lemma, clarify) 
+apply (rule minus_le_iff [THEN iffD1])  
+apply (rule_tac y=0 in order_trans, auto)
+done
+
+
+(* also spoilt by numeral arithmetic *)
+lemma cos_zero_iff: "(cos x = 0) =  
+      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |    
+       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
+apply (rule iffI)
+apply (cut_tac linorder_linear [of 0 x], safe)
+apply (drule cos_zero_lemma, assumption+)
+apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) 
+apply (force simp add: minus_equation_iff [of x]) 
+apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) 
+apply (auto simp add: cos_add)
+done
+
+(* ditto: but to a lesser extent *)
+lemma sin_zero_iff: "(sin x = 0) =  
+      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |    
+       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
+apply (rule iffI)
+apply (cut_tac linorder_linear [of 0 x], safe)
+apply (drule sin_zero_lemma, assumption+)
+apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
+apply (force simp add: minus_equation_iff [of x]) 
+apply (auto simp add: even_mult_two_ex)
+done
+
+
+subsection{*Tangent*}
+
+lemma tan_zero [simp]: "tan 0 = 0"
+by (simp add: tan_def)
+
+lemma tan_pi [simp]: "tan pi = 0"
+by (simp add: tan_def)
+
+lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
+by (simp add: tan_def)
+
+lemma tan_minus [simp]: "tan (-x) = - tan x"
+by (simp add: tan_def minus_mult_left)
+
+lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
+by (simp add: tan_def)
+
+lemma lemma_tan_add1: 
+      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
+        ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
+apply (unfold tan_def real_divide_def)
+apply (auto simp del: inverse_mult_distrib simp add: inverse_mult_distrib [symmetric] mult_ac)
+apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
+apply (auto simp del: inverse_mult_distrib simp add: mult_assoc left_diff_distrib cos_add)
+done
+
+lemma add_tan_eq: 
+      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
+       ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
+apply (unfold tan_def)
+apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
+apply (auto simp add: mult_assoc left_distrib)
+apply (simp (no_asm) add: sin_add)
+done
+
+lemma tan_add: "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]  
+      ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
+apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
+apply (simp add: tan_def)
+done
+
+lemma tan_double: "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]  
+      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
+apply (insert tan_add [of x x]) 
+apply (simp add: mult_2 [symmetric])  
+apply (auto simp add: numeral_2_eq_2)
+done
+
+lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
+apply (unfold tan_def real_divide_def)
+apply (auto intro!: sin_gt_zero2 cos_gt_zero_pi intro!: real_mult_order positive_imp_inverse_positive)
+done
+
+lemma tan_less_zero: 
+  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
+proof -
+  have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) 
+  thus ?thesis by simp
+qed
+
+lemma lemma_DERIV_tan:
+     "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
+apply (rule lemma_DERIV_subst)
+apply (best intro!: DERIV_intros intro: DERIV_chain2) 
+apply (auto simp add: real_divide_def numeral_2_eq_2)
+done
+
+lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
+by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
+
+lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
+apply (unfold real_divide_def)
+apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
+apply simp 
+apply (rule LIM_mult2)
+apply (rule_tac [2] inverse_1 [THEN subst])
+apply (rule_tac [2] LIM_inverse)
+apply (simp_all add: divide_inverse [symmetric]) 
+apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) 
+apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
+done
+
+lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
+apply (cut_tac LIM_cos_div_sin)
+apply (simp only: LIM_def)
+apply (drule_tac x = "inverse y" in spec, safe, force)
+apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
+apply (rule_tac x = " (pi/2) - e" in exI)
+apply (simp (no_asm_simp))
+apply (drule_tac x = " (pi/2) - e" in spec)
+apply (auto simp add: abs_eqI2 tan_def)
+apply (rule inverse_less_iff_less [THEN iffD1])
+apply (auto simp add: real_divide_def)
+apply (rule real_mult_order)
+apply (subgoal_tac [3] "0 < sin e")
+apply (subgoal_tac [3] "0 < cos e")
+apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: inverse_mult_distrib abs_mult)
+apply (drule_tac a = "cos e" in positive_imp_inverse_positive)
+apply (drule_tac x = "inverse (cos e) " in abs_eqI2)
+apply (auto dest!: abs_eqI2 simp add: mult_ac)
+done
+
+lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
+apply (frule real_le_imp_less_or_eq, safe)
+ prefer 2 apply force
+apply (drule lemma_tan_total, safe)
+apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
+apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
+apply (drule_tac y = xa in order_le_imp_less_or_eq)
+apply (auto dest: cos_gt_zero)
+done
+
+lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
+apply (cut_tac linorder_linear [of 0 y], safe)
+apply (drule tan_total_pos)
+apply (cut_tac [2] y="-y" in tan_total_pos, safe)
+apply (rule_tac [3] x = "-x" in exI)
+apply (auto intro!: exI)
+done
+
+lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
+apply (cut_tac y = y in lemma_tan_total1, auto)
+apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
+apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
+apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
+apply (rule_tac [4] Rolle)
+apply (rule_tac [2] Rolle)
+apply (auto intro!: DERIV_tan DERIV_isCont exI 
+            simp add: differentiable_def)
+txt{*Now, simulate TRYALL*}
+apply (rule_tac [!] DERIV_tan asm_rl)
+apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
+	    simp add: cos_gt_zero_pi [THEN real_not_refl2, THEN not_sym]) 
+done
+
+lemma arcsin_pi: "[| -1 \<le> y; y \<le> 1 |]  
+      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
+apply (drule sin_total, assumption)
+apply (erule ex1E)
+apply (unfold arcsin_def)
+apply (rule someI2, blast) 
+apply (force intro: order_trans) 
+done
+
+lemma arcsin: "[| -1 \<le> y; y \<le> 1 |]  
+      ==> -(pi/2) \<le> arcsin y &  
+           arcsin y \<le> pi/2 & sin(arcsin y) = y"
+apply (unfold arcsin_def)
+apply (drule sin_total, assumption)
+apply (fast intro: someI2)
+done
+
+lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
+by (blast dest: arcsin)
+      
+lemma arcsin_bounded:
+     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
+by (blast dest: arcsin)
+
+lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
+by (blast dest: arcsin)
+
+lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
+by (blast dest: arcsin)
+
+lemma arcsin_lt_bounded:
+     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
+apply (frule order_less_imp_le)
+apply (frule_tac y = y in order_less_imp_le)
+apply (frule arcsin_bounded)
+apply (safe, simp)
+apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
+apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
+apply (drule_tac [!] f = sin in arg_cong, auto)
+done
+
+lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
+apply (unfold arcsin_def)
+apply (rule some1_equality)
+apply (rule sin_total, auto)
+done
+
+lemma arcos: "[| -1 \<le> y; y \<le> 1 |]  
+      ==> 0 \<le> arcos y & arcos y \<le> pi & cos(arcos y) = y"
+apply (unfold arcos_def)
+apply (drule cos_total, assumption)
+apply (fast intro: someI2)
+done
+
+lemma cos_arcos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arcos y) = y"
+by (blast dest: arcos)
+      
+lemma arcos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y & arcos y \<le> pi"
+by (blast dest: arcos)
+
+lemma arcos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y"
+by (blast dest: arcos)
+
+lemma arcos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcos y \<le> pi"
+by (blast dest: arcos)
+
+lemma arcos_lt_bounded: "[| -1 < y; y < 1 |]  
+      ==> 0 < arcos y & arcos y < pi"
+apply (frule order_less_imp_le)
+apply (frule_tac y = y in order_less_imp_le)
+apply (frule arcos_bounded, auto)
+apply (drule_tac y = "arcos y" in order_le_imp_less_or_eq)
+apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
+apply (drule_tac [!] f = cos in arg_cong, auto)
+done
+
+lemma arcos_cos: "[|0 \<le> x; x \<le> pi |] ==> arcos(cos x) = x"
+apply (unfold arcos_def)
+apply (auto intro!: some1_equality cos_total)
+done
+
+lemma arcos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arcos(cos x) = -x"
+apply (unfold arcos_def)
+apply (auto intro!: some1_equality cos_total)
+done
+
+lemma arctan [simp]:
+     "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
+apply (cut_tac y = y in tan_total)
+apply (unfold arctan_def)
+apply (fast intro: someI2)
+done
+
+lemma tan_arctan: "tan(arctan y) = y"
+by auto
+
+lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
+by (auto simp only: arctan)
+
+lemma arctan_lbound: "- (pi/2) < arctan y"
+by auto
+
+lemma arctan_ubound: "arctan y < pi/2"
+by (auto simp only: arctan)
+
+lemma arctan_tan: 
+      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
+apply (unfold arctan_def)
+apply (rule some1_equality)
+apply (rule tan_total, auto)
+done
+
+lemma arctan_zero_zero [simp]: "arctan 0 = 0"
+by (insert arctan_tan [of 0], simp)
+
+lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
+apply (auto simp add: cos_zero_iff)
+apply (case_tac "n")
+apply (case_tac [3] "n")
+apply (cut_tac [2] y = x in arctan_ubound)
+apply (cut_tac [4] y = x in arctan_lbound) 
+apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
+done
+
+lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
+apply (rule power_inverse [THEN subst])
+apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
+apply (auto dest: realpow_not_zero 
+        simp add: power_mult_distrib left_distrib realpow_divide tan_def 
+                  mult_assoc power_inverse [symmetric] 
+        simp del: realpow_Suc)
+done
+
+lemma lemma_sin_cos_eq [simp]: "sin (xa + 1 / 2 * real (Suc m) * pi) =  
+      cos (xa + 1 / 2 * real  (m) * pi)"
+apply (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
+done
+
+lemma lemma_sin_cos_eq2 [simp]: "sin (xa + real (Suc m) * pi / 2) =  
+      cos (xa + real (m) * pi / 2)"
+apply (simp only: cos_add sin_add real_divide_def real_of_nat_Suc left_distrib right_distrib, auto)
+done
+
+lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
+apply (rule lemma_DERIV_subst)
+apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
+apply (best intro!: DERIV_intros intro: DERIV_chain2)+
+apply (simp (no_asm))
+done
+
+(* which further simplifies to (- 1 ^ m) !! *)
+lemma sin_cos_npi [simp]: "sin ((real m + 1/2) * pi) = cos (real m * pi)"
+by (auto simp add: right_distrib sin_add left_distrib mult_ac)
+
+lemma sin_cos_npi2 [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
+apply (cut_tac m = n in sin_cos_npi)
+apply (simp only: real_of_nat_Suc left_distrib real_divide_def, auto)
+done
+
+lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
+by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
+
+lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
+apply (subgoal_tac "3/2 = (1+1 / 2::real)")
+apply (simp only: left_distrib) 
+apply (auto simp add: cos_add mult_ac)
+done
+
+lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
+by (auto simp add: mult_assoc)
+
+lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
+apply (subgoal_tac "3/2 = (1+1 / 2::real)")
+apply (simp only: left_distrib) 
+apply (auto simp add: sin_add mult_ac)
+done
+
+(*NEEDED??*)
+lemma [simp]: "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
+apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
+done
+
+(*NEEDED??*)
+lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
+apply (simp only: cos_add sin_add real_divide_def real_of_nat_Suc left_distrib right_distrib, auto)
+done
+
+lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
+by (simp only: cos_add sin_add real_divide_def real_of_nat_Suc left_distrib right_distrib, auto)
+
+lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
+apply (rule lemma_DERIV_subst)
+apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
+apply (best intro!: DERIV_intros intro: DERIV_chain2)+
+apply (simp (no_asm))
+done
+
+lemma isCont_cos [simp]: "isCont cos x"
+by (rule DERIV_cos [THEN DERIV_isCont])
+
+lemma isCont_sin [simp]: "isCont sin x"
+by (rule DERIV_sin [THEN DERIV_isCont])
+
+lemma isCont_exp [simp]: "isCont exp x"
+by (rule DERIV_exp [THEN DERIV_isCont])
+
+lemma sin_zero_abs_cos_one: "sin x = 0 ==> abs(cos x) = 1"
+by (auto simp add: sin_zero_iff even_mult_two_ex)
+
+lemma exp_eq_one_iff [simp]: "(exp x = 1) = (x = 0)"
+apply auto
+apply (drule_tac f = ln in arg_cong, auto)
+done
+
+lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
+by (cut_tac x = x in sin_cos_squared_add3, auto)
+
+
+lemma real_root_less_mono: "[| 0 \<le> x; x < y |] ==> root(Suc n) x < root(Suc n) y"
+apply (frule order_le_less_trans, assumption)
+apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst])
+apply (rotate_tac 1, assumption)
+apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst])
+apply (rotate_tac 3, assumption)
+apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le)
+apply (frule_tac n = n in real_root_pos_pos_le)
+apply (frule_tac n = n in real_root_pos_pos)
+apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing)
+apply (assumption, assumption)
+apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq)
+apply auto
+apply (drule_tac f = "%x. x ^ (Suc n) " in arg_cong)
+apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc)
+done
+
+lemma real_root_le_mono: "[| 0 \<le> x; x \<le> y |] ==> root(Suc n) x \<le> root(Suc n) y"
+apply (drule_tac y = y in order_le_imp_less_or_eq)
+apply (auto dest: real_root_less_mono intro: order_less_imp_le)
+done
+
+lemma real_root_less_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)"
+apply (auto intro: real_root_less_mono)
+apply (rule ccontr, drule linorder_not_less [THEN iffD1])
+apply (drule_tac x = y and n = n in real_root_le_mono, auto)
+done
+
+lemma real_root_le_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x \<le> root(Suc n) y) = (x \<le> y)"
+apply (auto intro: real_root_le_mono)
+apply (simp (no_asm) add: linorder_not_less [symmetric])
+apply auto
+apply (drule_tac x = y and n = n in real_root_less_mono, auto)
+done
+
+lemma real_root_eq_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)"
+apply (auto intro!: order_antisym)
+apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1])
+apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto)
+done
+
+lemma real_root_pos_unique: "[| 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x |] ==> root (Suc n) x = y"
+by (auto dest: real_root_pos2 simp del: realpow_Suc)
+
+lemma real_root_mult: "[| 0 \<le> x; 0 \<le> y |] 
+      ==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y"
+apply (rule real_root_pos_unique)
+apply (auto intro!: real_root_pos_pos_le simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2 simp del: realpow_Suc)
+done
+
+lemma real_root_inverse: "0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))"
+apply (rule real_root_pos_unique)
+apply (auto intro: real_root_pos_pos_le simp add: power_inverse [symmetric] real_root_pow_pos2 simp del: realpow_Suc)
+done
+
+lemma real_root_divide: 
+     "[| 0 \<le> x; 0 \<le> y |]  
+      ==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)"
+apply (unfold real_divide_def)
+apply (auto simp add: real_root_mult real_root_inverse)
+done
+
+lemma real_sqrt_less_mono: "[| 0 \<le> x; x < y |] ==> sqrt(x) < sqrt(y)"
+apply (unfold sqrt_def)
+apply (auto intro: real_root_less_mono)
+done
+
+lemma real_sqrt_le_mono: "[| 0 \<le> x; x \<le> y |] ==> sqrt(x) \<le> sqrt(y)"
+apply (unfold sqrt_def)
+apply (auto intro: real_root_le_mono)
+done
+
+lemma real_sqrt_less_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) < sqrt(y)) = (x < y)"
+by (unfold sqrt_def, auto)
+
+lemma real_sqrt_le_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) \<le> sqrt(y)) = (x \<le> y)"
+by (unfold sqrt_def, auto)
+
+lemma real_sqrt_eq_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) = sqrt(y)) = (x = y)"
+by (unfold sqrt_def, auto)
+
+lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)"
+apply (rule real_sqrt_one [THEN subst], safe)
+apply (rule_tac [2] real_sqrt_less_mono)
+apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto)
+done
+
+lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)"
+apply (rule real_sqrt_one [THEN subst], safe)
+apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto)
+done
+
+lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
+apply (unfold real_divide_def)
+apply (case_tac "r=0")
+apply (auto simp add: inverse_mult_distrib mult_ac)
+done
+
+
+subsection{*Theorems About Sqrt, Transcendental Functions for Complex*}
+
+lemma lemma_real_divide_sqrt: 
+    "0 < x ==> 0 \<le> x/(sqrt (x * x + y * y))"
+apply (unfold real_divide_def)
+apply (rule real_mult_order [THEN order_less_imp_le], assumption)
+apply (subgoal_tac "0 < inverse (sqrt (x\<twosuperior> + y\<twosuperior>))") 
+ apply (simp add: numeral_2_eq_2)
+apply (simp add: real_sqrt_sum_squares_ge1 [THEN [2] order_less_le_trans]) 
+done
+
+lemma lemma_real_divide_sqrt_ge_minus_one:
+     "0 < x ==> -1 \<le> x/(sqrt (x * x + y * y))"
+apply (rule real_le_trans)
+apply (rule_tac [2] lemma_real_divide_sqrt, auto)
+done
+
+lemma real_sqrt_sum_squares_gt_zero1: "x < 0 ==> 0 < sqrt (x * x + y * y)"
+apply (rule real_sqrt_gt_zero)
+apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
+apply (auto simp add: zero_less_mult_iff)
+done
+
+lemma real_sqrt_sum_squares_gt_zero2: "0 < x ==> 0 < sqrt (x * x + y * y)"
+apply (rule real_sqrt_gt_zero)
+apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
+apply (auto simp add: zero_less_mult_iff)
+done
+
+lemma real_sqrt_sum_squares_gt_zero3: "x \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
+apply (cut_tac x = x and y = 0 in linorder_less_linear)
+apply (auto intro: real_sqrt_sum_squares_gt_zero2 real_sqrt_sum_squares_gt_zero1 simp add: numeral_2_eq_2)
+done
+
+lemma real_sqrt_sum_squares_gt_zero3a: "y \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
+apply (drule_tac y = x in real_sqrt_sum_squares_gt_zero3)
+apply (auto simp add: real_add_commute)
+done
+
+lemma real_sqrt_sum_squares_eq_cancel [simp]: "sqrt(x\<twosuperior> + y\<twosuperior>) = x ==> y = 0"
+by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, auto)
+
+lemma real_sqrt_sum_squares_eq_cancel2 [simp]: "sqrt(x\<twosuperior> + y\<twosuperior>) = y ==> x = 0"
+apply (rule_tac x = y in real_sqrt_sum_squares_eq_cancel)
+apply (simp add: real_add_commute)
+done
+
+lemma lemma_real_divide_sqrt_le_one: "x < 0 ==> x/(sqrt (x * x + y * y)) \<le> 1"
+by (insert lemma_real_divide_sqrt_ge_minus_one [of "-x" y], simp)
+
+lemma lemma_real_divide_sqrt_ge_minus_one2:
+     "x < 0 ==> -1 \<le> x/(sqrt (x * x + y * y))"
+apply (case_tac "y = 0", auto)
+apply (frule abs_minus_eqI2)
+apply (auto simp add: inverse_minus_eq)
+apply (rule order_less_imp_le)
+apply (rule_tac z1 = "sqrt (x * x + y * y) " in real_mult_less_iff1 [THEN iffD1])
+apply (frule_tac [2] y2 = y in
+       real_sqrt_sum_squares_gt_zero1 [THEN real_not_refl2, THEN not_sym])
+apply (auto intro: real_sqrt_sum_squares_gt_zero1 simp add: mult_ac)
+apply (cut_tac x = "-x" and y = y in real_sqrt_sum_squares_ge1)
+apply (drule order_le_less [THEN iffD1], safe) 
+apply (simp add: numeral_2_eq_2)
+apply (drule sym [THEN real_sqrt_sum_squares_eq_cancel], simp)
+done
+
+lemma lemma_real_divide_sqrt_le_one2: "0 < x ==> x/(sqrt (x * x + y * y)) \<le> 1"
+by (cut_tac x = "-x" and y = y in lemma_real_divide_sqrt_ge_minus_one2, auto)
+
+
+lemma cos_x_y_ge_minus_one: "-1 \<le> x / sqrt (x * x + y * y)"
+apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
+apply (rule lemma_real_divide_sqrt_ge_minus_one2)
+apply (rule_tac [3] lemma_real_divide_sqrt_ge_minus_one, auto)
+done
+
+lemma cos_x_y_ge_minus_one1a [simp]: "-1 \<le> y / sqrt (x * x + y * y)"
+apply (cut_tac x = y and y = x in cos_x_y_ge_minus_one)
+apply (simp add: real_add_commute)
+done
+
+lemma cos_x_y_le_one [simp]: "x / sqrt (x * x + y * y) \<le> 1"
+apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
+apply (rule lemma_real_divide_sqrt_le_one)
+apply (rule_tac [3] lemma_real_divide_sqrt_le_one2, auto)
+done
+
+lemma cos_x_y_le_one2 [simp]: "y / sqrt (x * x + y * y) \<le> 1"
+apply (cut_tac x = y and y = x in cos_x_y_le_one)
+apply (simp add: real_add_commute)
+done
+
+declare cos_arcos [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
+declare arcos_bounded [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
+
+declare cos_arcos [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
+declare arcos_bounded [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
+
+lemma cos_abs_x_y_ge_minus_one [simp]:
+     "-1 \<le> \<bar>x\<bar> / sqrt (x * x + y * y)"
+apply (cut_tac x = x and y = 0 in linorder_less_linear)
+apply (auto simp add: abs_minus_eqI2 abs_eqI2)
+apply (drule lemma_real_divide_sqrt_ge_minus_one, force)
+done
+
+lemma cos_abs_x_y_le_one [simp]: "\<bar>x\<bar> / sqrt (x * x + y * y) \<le> 1"
+apply (cut_tac x = x and y = 0 in linorder_less_linear)
+apply (auto simp add: abs_minus_eqI2 abs_eqI2)
+apply (drule lemma_real_divide_sqrt_ge_minus_one2, force)
+done
+
+declare cos_arcos [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
+declare arcos_bounded [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
+
+lemma minus_pi_less_zero: "-pi < 0"
+apply (simp (no_asm))
+done
+declare minus_pi_less_zero [simp]
+declare minus_pi_less_zero [THEN order_less_imp_le, simp]
+
+lemma arcos_ge_minus_pi: "[| -1 \<le> y; y \<le> 1 |] ==> -pi \<le> arcos y"
+apply (rule real_le_trans)
+apply (rule_tac [2] arcos_lbound, auto)
+done
+
+declare arcos_ge_minus_pi [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
+
+(* How tedious! *)
+lemma lemma_divide_rearrange:
+     "[| x + (y::real) \<noteq> 0; 1 - z = x/(x + y) |] ==> z = y/(x + y)"
+apply (rule_tac c1 = "x + y" in real_mult_right_cancel [THEN iffD1])
+apply (frule_tac [2] c1 = "x + y" in real_mult_right_cancel [THEN iffD2])
+prefer 2 apply assumption
+apply (rotate_tac [2] 2)
+apply (drule_tac [2] mult_assoc [THEN subst])
+apply (rotate_tac [2] 2)
+apply (frule_tac [2] left_inverse [THEN subst])
+prefer 2 apply assumption
+apply (erule_tac [2] V = " (1 - z) * (x + y) = x / (x + y) * (x + y) " in thin_rl)
+apply (erule_tac [2] V = "1 - z = x / (x + y) " in thin_rl)
+apply (auto simp add: mult_assoc)
+apply (auto simp add: right_distrib left_diff_distrib)
+done
+
+lemma lemma_cos_sin_eq:
+     "[| 0 < x * x + y * y;  
+         1 - (sin xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2 |] 
+      ==> (sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2"
+by (auto intro: lemma_divide_rearrange
+         simp add: realpow_divide power2_eq_square [symmetric])
+
+
+lemma lemma_sin_cos_eq:
+     "[| 0 < x * x + y * y;  
+         1 - (cos xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2 |]
+      ==> (cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2"
+apply (auto simp add: realpow_divide power2_eq_square [symmetric])
+apply (rule add_commute [THEN subst])
+apply (rule lemma_divide_rearrange, simp)
+apply (simp add: add_commute)
+done
+
+lemma sin_x_y_disj:
+     "[| x \<noteq> 0;  
+         cos xa = x / sqrt (x * x + y * y)  
+      |] ==>  sin xa = y / sqrt (x * x + y * y) |  
+              sin xa = - y / sqrt (x * x + y * y)"
+apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
+apply (frule_tac y = y in real_sum_square_gt_zero)
+apply (simp add: cos_squared_eq)
+apply (subgoal_tac "(sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2")
+apply (rule_tac [2] lemma_cos_sin_eq)
+apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
+done
+
+lemma lemma_cos_not_eq_zero: "x \<noteq> 0 ==> x / sqrt (x * x + y * y) \<noteq> 0"
+apply (unfold real_divide_def)
+apply (frule_tac y3 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym, THEN nonzero_imp_inverse_nonzero])
+apply (auto simp add: power2_eq_square)
+done
+
+lemma cos_x_y_disj: "[| x \<noteq> 0;  
+         sin xa = y / sqrt (x * x + y * y)  
+      |] ==>  cos xa = x / sqrt (x * x + y * y) |  
+              cos xa = - x / sqrt (x * x + y * y)"
+apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
+apply (frule_tac y = y in real_sum_square_gt_zero)
+apply (simp add: sin_squared_eq del: realpow_Suc)
+apply (subgoal_tac "(cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2")
+apply (rule_tac [2] lemma_sin_cos_eq)
+apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
+done
+
+lemma real_sqrt_divide_less_zero: "0 < y ==> - y / sqrt (x * x + y * y) < 0"
+apply (case_tac "x = 0")
+apply (auto simp add: abs_eqI2)
+apply (drule_tac y = y in real_sqrt_sum_squares_gt_zero3)
+apply (auto simp add: zero_less_mult_iff real_divide_def power2_eq_square)
+done
+
+lemma polar_ex1: "[| x \<noteq> 0; 0 < y |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
+apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>) " in exI)
+apply (rule_tac x = "arcos (x / sqrt (x * x + y * y))" in exI)
+apply auto
+apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
+apply (auto simp add: power2_eq_square)
+apply (unfold arcos_def)
+apply (cut_tac x1 = x and y1 = y 
+       in cos_total [OF cos_x_y_ge_minus_one cos_x_y_le_one])
+apply (rule someI2_ex, blast)
+apply (erule_tac V = "EX! xa. 0 \<le> xa & xa \<le> pi & cos xa = x / sqrt (x * x + y * y) " in thin_rl)
+apply (frule sin_x_y_disj, blast)
+apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
+apply (auto simp add: power2_eq_square)
+apply (drule sin_ge_zero, assumption)
+apply (drule_tac x = x in real_sqrt_divide_less_zero, auto)
+done
+
+lemma real_sum_squares_cancel2a: "x * x = -(y * y) ==> y = (0::real)"
+by (auto intro: real_sum_squares_cancel)
+
+lemma polar_ex2: "[| x \<noteq> 0; y < 0 |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
+apply (cut_tac x = 0 and y = x in linorder_less_linear, auto)
+apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>) " in exI)
+apply (rule_tac x = "arcsin (y / sqrt (x * x + y * y))" in exI)
+apply (auto dest: real_sum_squares_cancel2a simp add: power2_eq_square)
+apply (unfold arcsin_def)
+apply (cut_tac x1 = x and y1 = y 
+       in sin_total [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2])
+apply (rule someI2_ex, blast)
+apply (erule_tac V = "EX! xa. - (pi/2) \<le> xa & xa \<le> pi/2 & sin xa = y / sqrt (x * x + y * y) " in thin_rl)
+apply (cut_tac x=x and y=y in cos_x_y_disj, simp, blast, auto)
+apply (drule cos_ge_zero, force)
+apply (drule_tac x = y in real_sqrt_divide_less_zero)
+apply (auto simp add: real_add_commute)
+apply (insert polar_ex1 [of x "-y"], simp, clarify) 
+apply (rule_tac x = r in exI)
+apply (rule_tac x = "-a" in exI, simp) 
+done
+
+lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
+apply (case_tac "x = 0", auto)
+apply (rule_tac x = y in exI)
+apply (rule_tac x = "pi/2" in exI, auto)
+apply (cut_tac x = 0 and y = y in linorder_less_linear, auto)
+apply (rule_tac [2] x = x in exI)
+apply (rule_tac [2] x = 0 in exI, auto)
+apply (blast intro: polar_ex1 polar_ex2)+
+done
+
+lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
+apply (rule_tac n = 1 in realpow_increasing)
+apply (auto simp add: numeral_2_eq_2 [symmetric])
+done
+
+lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
+apply (rule real_add_commute [THEN subst])
+apply (rule real_sqrt_ge_abs1)
+done
+declare real_sqrt_ge_abs1 [simp] real_sqrt_ge_abs2 [simp]
+
+lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
+by (auto intro: real_sqrt_gt_zero)
+
+lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2"
+by (auto intro: real_sqrt_ge_zero)
+
+lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
+apply (rule order_less_le_trans [of _ "7/5"], simp) 
+apply (rule_tac n = 1 in realpow_increasing)
+  prefer 3 apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
+apply (simp_all add: numeral_2_eq_2)
+done
+
+lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
+apply (rule_tac z1 = "inverse u" in real_mult_less_iff1 [THEN iffD1], auto)
+apply (rule_tac z1 = "sqrt 2" in real_mult_less_iff1 [THEN iffD1], auto)
+done
+
+lemma four_x_squared: 
+  fixes x::real
+  shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>"
+by (simp add: power2_eq_square)
+
+
+text{*Needed for the infinitely close relation over the nonstandard
+    complex numbers*}
+lemma lemma_sqrt_hcomplex_capprox:
+     "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u"
+apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
+apply (erule_tac [2] lemma_real_divide_sqrt_less)
+apply (rule_tac n = 1 in realpow_increasing)
+apply (auto simp add: real_0_le_divide_iff realpow_divide numeral_2_eq_2 [symmetric] 
+        simp del: realpow_Suc)
+apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst])
+apply (rule add_mono)
+apply (auto simp add: four_x_squared simp del: realpow_Suc intro: power_mono)
+done
+
+declare real_sqrt_sum_squares_ge_zero [THEN abs_eqI1, simp]
+
+
+subsection{*A Few Theorems Involving Ln, Derivatives, etc.*}
+
+lemma lemma_DERIV_ln:
+     "DERIV ln z :> l ==> DERIV (%x. exp (ln x)) z :> exp (ln z) * l"
+by (erule DERIV_fun_exp)
+
+lemma STAR_exp_ln: "0 < z ==> ( *f* (%x. exp (ln x))) z = z"
+apply (rule_tac z = z in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_zero_def hypreal_less)
+done
+
+lemma hypreal_add_Infinitesimal_gt_zero: "[|e : Infinitesimal; 0 < x |] ==> 0 < hypreal_of_real x + e"
+apply (rule_tac c1 = "-e" in add_less_cancel_right [THEN iffD1])
+apply (auto intro: Infinitesimal_less_SReal)
+done
+
+lemma NSDERIV_exp_ln_one: "0 < z ==> NSDERIV (%x. exp (ln x)) z :> 1"
+apply (unfold nsderiv_def NSLIM_def, auto)
+apply (rule ccontr)
+apply (subgoal_tac "0 < hypreal_of_real z + h")
+apply (drule STAR_exp_ln)
+apply (rule_tac [2] hypreal_add_Infinitesimal_gt_zero)
+apply (subgoal_tac "h/h = 1")
+apply (auto simp add: exp_ln_iff [symmetric] simp del: exp_ln_iff)
+done
+
+lemma DERIV_exp_ln_one: "0 < z ==> DERIV (%x. exp (ln x)) z :> 1"
+by (auto intro: NSDERIV_exp_ln_one simp add: NSDERIV_DERIV_iff [symmetric])
+
+lemma lemma_DERIV_ln2: "[| 0 < z; DERIV ln z :> l |] ==>  exp (ln z) * l = 1"
+apply (rule DERIV_unique)
+apply (rule lemma_DERIV_ln)
+apply (rule_tac [2] DERIV_exp_ln_one, auto)
+done
+
+lemma lemma_DERIV_ln3: "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/(exp (ln z))"
+apply (rule_tac c1 = "exp (ln z) " in real_mult_left_cancel [THEN iffD1])
+apply (auto intro: lemma_DERIV_ln2)
+done
+
+lemma lemma_DERIV_ln4: "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/z"
+apply (rule_tac t = z in exp_ln_iff [THEN iffD2, THEN subst])
+apply (auto intro: lemma_DERIV_ln3)
+done
+
+(* need to rename second isCont_inverse *)
+
+lemma isCont_inv_fun: "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
+         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
+      ==> isCont g (f x)"
+apply (simp (no_asm) add: isCont_iff LIM_def)
+apply safe
+apply (drule_tac ?d1.0 = r in real_lbound_gt_zero)
+apply (assumption, safe)
+apply (subgoal_tac "\<forall>z. \<bar>z - x\<bar> \<le> e --> (g (f z) = z) ")
+prefer 2 apply force
+apply (subgoal_tac "\<forall>z. \<bar>z - x\<bar> \<le> e --> isCont f z")
+prefer 2 apply force
+apply (drule_tac d = e in isCont_inj_range)
+prefer 2 apply (assumption, assumption, safe)
+apply (rule_tac x = ea in exI, auto)
+apply (rotate_tac 4)
+apply (drule_tac x = "f (x) + xa" in spec)
+apply auto
+apply (drule sym, auto, arith)
+done
+
+lemma isCont_inv_fun_inv:
+     "[| 0 < d;  
+         \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
+         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
+       ==> \<exists>e. 0 < e &  
+             (\<forall>y. 0 < abs(y - f(x)) & abs(y - f(x)) < e --> f(g(y)) = y)"
+apply (drule isCont_inj_range)
+prefer 2 apply (assumption, assumption, auto)
+apply (rule_tac x = e in exI, auto)
+apply (rotate_tac 2)
+apply (drule_tac x = y in spec, auto)
+done
+
+
+text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
+lemma LIM_fun_gt_zero: "[| f -- c --> l; 0 < l |]  
+         ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
+apply (auto simp add: LIM_def)
+apply (drule_tac x = "l/2" in spec, safe, force)
+apply (rule_tac x = s in exI)
+apply (auto simp only: abs_interval_iff)
+done
+
+lemma LIM_fun_less_zero: "[| f -- c --> l; l < 0 |]  
+         ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
+apply (auto simp add: LIM_def)
+apply (drule_tac x = "-l/2" in spec, safe, force)
+apply (rule_tac x = s in exI)
+apply (auto simp only: abs_interval_iff)
+done
+
+
+lemma LIM_fun_not_zero:
+     "[| f -- c --> l; l \<noteq> 0 |] 
+      ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
+apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
+apply (drule LIM_fun_less_zero)
+apply (drule_tac [3] LIM_fun_gt_zero, auto)
+apply (rule_tac [!] x = r in exI, auto)
+done
+
+ML
+{*
+val inverse_unique = thm "inverse_unique";
+val real_root_zero = thm "real_root_zero";
+val real_root_pow_pos = thm "real_root_pow_pos";
+val real_root_pow_pos2 = thm "real_root_pow_pos2";
+val real_root_pos = thm "real_root_pos";
+val real_root_pos2 = thm "real_root_pos2";
+val real_root_pos_pos = thm "real_root_pos_pos";
+val real_root_pos_pos_le = thm "real_root_pos_pos_le";
+val real_root_one = thm "real_root_one";
+val root_2_eq = thm "root_2_eq";
+val real_sqrt_zero = thm "real_sqrt_zero";
+val real_sqrt_one = thm "real_sqrt_one";
+val real_sqrt_pow2_iff = thm "real_sqrt_pow2_iff";
+val real_sqrt_pow2 = thm "real_sqrt_pow2";
+val real_sqrt_abs_abs = thm "real_sqrt_abs_abs";
+val real_pow_sqrt_eq_sqrt_pow = thm "real_pow_sqrt_eq_sqrt_pow";
+val real_pow_sqrt_eq_sqrt_abs_pow2 = thm "real_pow_sqrt_eq_sqrt_abs_pow2";
+val real_sqrt_pow_abs = thm "real_sqrt_pow_abs";
+val not_real_square_gt_zero = thm "not_real_square_gt_zero";
+val real_mult_self_eq_zero_iff = thm "real_mult_self_eq_zero_iff";
+val real_sqrt_gt_zero = thm "real_sqrt_gt_zero";
+val real_sqrt_ge_zero = thm "real_sqrt_ge_zero";
+val sqrt_eqI = thm "sqrt_eqI";
+val real_sqrt_mult_distrib = thm "real_sqrt_mult_distrib";
+val real_sqrt_mult_distrib2 = thm "real_sqrt_mult_distrib2";
+val real_sqrt_sum_squares_ge_zero = thm "real_sqrt_sum_squares_ge_zero";
+val real_sqrt_sum_squares_mult_ge_zero = thm "real_sqrt_sum_squares_mult_ge_zero";
+val real_sqrt_sum_squares_mult_squared_eq = thm "real_sqrt_sum_squares_mult_squared_eq";
+val real_sqrt_abs = thm "real_sqrt_abs";
+val real_sqrt_abs2 = thm "real_sqrt_abs2";
+val real_sqrt_pow2_gt_zero = thm "real_sqrt_pow2_gt_zero";
+val real_sqrt_not_eq_zero = thm "real_sqrt_not_eq_zero";
+val real_inv_sqrt_pow2 = thm "real_inv_sqrt_pow2";
+val real_sqrt_eq_zero_cancel = thm "real_sqrt_eq_zero_cancel";
+val real_sqrt_eq_zero_cancel_iff = thm "real_sqrt_eq_zero_cancel_iff";
+val real_sqrt_sum_squares_ge1 = thm "real_sqrt_sum_squares_ge1";
+val real_sqrt_sum_squares_ge2 = thm "real_sqrt_sum_squares_ge2";
+val real_sqrt_ge_one = thm "real_sqrt_ge_one";
+val summable_exp = thm "summable_exp";
+val summable_sin = thm "summable_sin";
+val summable_cos = thm "summable_cos";
+val exp_converges = thm "exp_converges";
+val sin_converges = thm "sin_converges";
+val cos_converges = thm "cos_converges";
+val powser_insidea = thm "powser_insidea";
+val powser_inside = thm "powser_inside";
+val diffs_minus = thm "diffs_minus";
+val diffs_equiv = thm "diffs_equiv";
+val less_add_one = thm "less_add_one";
+val termdiffs_aux = thm "termdiffs_aux";
+val termdiffs = thm "termdiffs";
+val exp_fdiffs = thm "exp_fdiffs";
+val sin_fdiffs = thm "sin_fdiffs";
+val sin_fdiffs2 = thm "sin_fdiffs2";
+val cos_fdiffs = thm "cos_fdiffs";
+val cos_fdiffs2 = thm "cos_fdiffs2";
+val DERIV_exp = thm "DERIV_exp";
+val DERIV_sin = thm "DERIV_sin";
+val DERIV_cos = thm "DERIV_cos";
+val exp_zero = thm "exp_zero";
+val exp_ge_add_one_self = thm "exp_ge_add_one_self";
+val exp_gt_one = thm "exp_gt_one";
+val DERIV_exp_add_const = thm "DERIV_exp_add_const";
+val DERIV_exp_minus = thm "DERIV_exp_minus";
+val DERIV_exp_exp_zero = thm "DERIV_exp_exp_zero";
+val exp_add_mult_minus = thm "exp_add_mult_minus";
+val exp_mult_minus = thm "exp_mult_minus";
+val exp_mult_minus2 = thm "exp_mult_minus2";
+val exp_minus = thm "exp_minus";
+val exp_add = thm "exp_add";
+val exp_ge_zero = thm "exp_ge_zero";
+val exp_not_eq_zero = thm "exp_not_eq_zero";
+val exp_gt_zero = thm "exp_gt_zero";
+val inv_exp_gt_zero = thm "inv_exp_gt_zero";
+val abs_exp_cancel = thm "abs_exp_cancel";
+val exp_real_of_nat_mult = thm "exp_real_of_nat_mult";
+val exp_diff = thm "exp_diff";
+val exp_less_mono = thm "exp_less_mono";
+val exp_less_cancel = thm "exp_less_cancel";
+val exp_less_cancel_iff = thm "exp_less_cancel_iff";
+val exp_le_cancel_iff = thm "exp_le_cancel_iff";
+val exp_inj_iff = thm "exp_inj_iff";
+val exp_total = thm "exp_total";
+val ln_exp = thm "ln_exp";
+val exp_ln_iff = thm "exp_ln_iff";
+val ln_mult = thm "ln_mult";
+val ln_inj_iff = thm "ln_inj_iff";
+val ln_one = thm "ln_one";
+val ln_inverse = thm "ln_inverse";
+val ln_div = thm "ln_div";
+val ln_less_cancel_iff = thm "ln_less_cancel_iff";
+val ln_le_cancel_iff = thm "ln_le_cancel_iff";
+val ln_realpow = thm "ln_realpow";
+val ln_add_one_self_le_self = thm "ln_add_one_self_le_self";
+val ln_less_self = thm "ln_less_self";
+val ln_ge_zero = thm "ln_ge_zero";
+val ln_gt_zero = thm "ln_gt_zero";
+val ln_not_eq_zero = thm "ln_not_eq_zero";
+val ln_less_zero = thm "ln_less_zero";
+val exp_ln_eq = thm "exp_ln_eq";
+val sin_zero = thm "sin_zero";
+val cos_zero = thm "cos_zero";
+val DERIV_sin_sin_mult = thm "DERIV_sin_sin_mult";
+val DERIV_sin_sin_mult2 = thm "DERIV_sin_sin_mult2";
+val DERIV_sin_realpow2 = thm "DERIV_sin_realpow2";
+val DERIV_sin_realpow2a = thm "DERIV_sin_realpow2a";
+val DERIV_cos_cos_mult = thm "DERIV_cos_cos_mult";
+val DERIV_cos_cos_mult2 = thm "DERIV_cos_cos_mult2";
+val DERIV_cos_realpow2 = thm "DERIV_cos_realpow2";
+val DERIV_cos_realpow2a = thm "DERIV_cos_realpow2a";
+val DERIV_cos_realpow2b = thm "DERIV_cos_realpow2b";
+val DERIV_cos_cos_mult3 = thm "DERIV_cos_cos_mult3";
+val DERIV_sin_circle_all = thm "DERIV_sin_circle_all";
+val DERIV_sin_circle_all_zero = thm "DERIV_sin_circle_all_zero";
+val sin_cos_squared_add = thm "sin_cos_squared_add";
+val sin_cos_squared_add2 = thm "sin_cos_squared_add2";
+val sin_cos_squared_add3 = thm "sin_cos_squared_add3";
+val sin_squared_eq = thm "sin_squared_eq";
+val cos_squared_eq = thm "cos_squared_eq";
+val real_gt_one_ge_zero_add_less = thm "real_gt_one_ge_zero_add_less";
+val abs_sin_le_one = thm "abs_sin_le_one";
+val sin_ge_minus_one = thm "sin_ge_minus_one";
+val sin_le_one = thm "sin_le_one";
+val abs_cos_le_one = thm "abs_cos_le_one";
+val cos_ge_minus_one = thm "cos_ge_minus_one";
+val cos_le_one = thm "cos_le_one";
+val DERIV_fun_pow = thm "DERIV_fun_pow";
+val DERIV_fun_exp = thm "DERIV_fun_exp";
+val DERIV_fun_sin = thm "DERIV_fun_sin";
+val DERIV_fun_cos = thm "DERIV_fun_cos";
+val DERIV_intros = thms "DERIV_intros";
+val sin_cos_add = thm "sin_cos_add";
+val sin_add = thm "sin_add";
+val cos_add = thm "cos_add";
+val sin_cos_minus = thm "sin_cos_minus";
+val sin_minus = thm "sin_minus";
+val cos_minus = thm "cos_minus";
+val sin_diff = thm "sin_diff";
+val sin_diff2 = thm "sin_diff2";
+val cos_diff = thm "cos_diff";
+val cos_diff2 = thm "cos_diff2";
+val sin_double = thm "sin_double";
+val cos_double = thm "cos_double";
+val sin_paired = thm "sin_paired";
+val sin_gt_zero = thm "sin_gt_zero";
+val sin_gt_zero1 = thm "sin_gt_zero1";
+val cos_double_less_one = thm "cos_double_less_one";
+val cos_paired = thm "cos_paired";
+val cos_two_less_zero = thm "cos_two_less_zero";
+val cos_is_zero = thm "cos_is_zero";
+val pi_half = thm "pi_half";
+val cos_pi_half = thm "cos_pi_half";
+val pi_half_gt_zero = thm "pi_half_gt_zero";
+val pi_half_less_two = thm "pi_half_less_two";
+val pi_gt_zero = thm "pi_gt_zero";
+val pi_neq_zero = thm "pi_neq_zero";
+val pi_not_less_zero = thm "pi_not_less_zero";
+val pi_ge_zero = thm "pi_ge_zero";
+val minus_pi_half_less_zero = thm "minus_pi_half_less_zero";
+val sin_pi_half = thm "sin_pi_half";
+val cos_pi = thm "cos_pi";
+val sin_pi = thm "sin_pi";
+val sin_cos_eq = thm "sin_cos_eq";
+val minus_sin_cos_eq = thm "minus_sin_cos_eq";
+val cos_sin_eq = thm "cos_sin_eq";
+val sin_periodic_pi = thm "sin_periodic_pi";
+val sin_periodic_pi2 = thm "sin_periodic_pi2";
+val cos_periodic_pi = thm "cos_periodic_pi";
+val sin_periodic = thm "sin_periodic";
+val cos_periodic = thm "cos_periodic";
+val cos_npi = thm "cos_npi";
+val sin_npi = thm "sin_npi";
+val sin_npi2 = thm "sin_npi2";
+val cos_two_pi = thm "cos_two_pi";
+val sin_two_pi = thm "sin_two_pi";
+val sin_gt_zero2 = thm "sin_gt_zero2";
+val sin_less_zero = thm "sin_less_zero";
+val pi_less_4 = thm "pi_less_4";
+val cos_gt_zero = thm "cos_gt_zero";
+val cos_gt_zero_pi = thm "cos_gt_zero_pi";
+val cos_ge_zero = thm "cos_ge_zero";
+val sin_gt_zero_pi = thm "sin_gt_zero_pi";
+val sin_ge_zero = thm "sin_ge_zero";
+val cos_total = thm "cos_total";
+val sin_total = thm "sin_total";
+val reals_Archimedean4 = thm "reals_Archimedean4";
+val cos_zero_lemma = thm "cos_zero_lemma";
+val sin_zero_lemma = thm "sin_zero_lemma";
+val cos_zero_iff = thm "cos_zero_iff";
+val sin_zero_iff = thm "sin_zero_iff";
+val tan_zero = thm "tan_zero";
+val tan_pi = thm "tan_pi";
+val tan_npi = thm "tan_npi";
+val tan_minus = thm "tan_minus";
+val tan_periodic = thm "tan_periodic";
+val add_tan_eq = thm "add_tan_eq";
+val tan_add = thm "tan_add";
+val tan_double = thm "tan_double";
+val tan_gt_zero = thm "tan_gt_zero";
+val tan_less_zero = thm "tan_less_zero";
+val DERIV_tan = thm "DERIV_tan";
+val LIM_cos_div_sin = thm "LIM_cos_div_sin";
+val tan_total_pos = thm "tan_total_pos";
+val tan_total = thm "tan_total";
+val arcsin_pi = thm "arcsin_pi";
+val arcsin = thm "arcsin";
+val sin_arcsin = thm "sin_arcsin";
+val arcsin_bounded = thm "arcsin_bounded";
+val arcsin_lbound = thm "arcsin_lbound";
+val arcsin_ubound = thm "arcsin_ubound";
+val arcsin_lt_bounded = thm "arcsin_lt_bounded";
+val arcsin_sin = thm "arcsin_sin";
+val arcos = thm "arcos";
+val cos_arcos = thm "cos_arcos";
+val arcos_bounded = thm "arcos_bounded";
+val arcos_lbound = thm "arcos_lbound";
+val arcos_ubound = thm "arcos_ubound";
+val arcos_lt_bounded = thm "arcos_lt_bounded";
+val arcos_cos = thm "arcos_cos";
+val arcos_cos2 = thm "arcos_cos2";
+val arctan = thm "arctan";
+val tan_arctan = thm "tan_arctan";
+val arctan_bounded = thm "arctan_bounded";
+val arctan_lbound = thm "arctan_lbound";
+val arctan_ubound = thm "arctan_ubound";
+val arctan_tan = thm "arctan_tan";
+val arctan_zero_zero = thm "arctan_zero_zero";
+val cos_arctan_not_zero = thm "cos_arctan_not_zero";
+val tan_sec = thm "tan_sec";
+val DERIV_sin_add = thm "DERIV_sin_add";
+val sin_cos_npi = thm "sin_cos_npi";
+val sin_cos_npi2 = thm "sin_cos_npi2";
+val cos_2npi = thm "cos_2npi";
+val cos_3over2_pi = thm "cos_3over2_pi";
+val sin_2npi = thm "sin_2npi";
+val sin_3over2_pi = thm "sin_3over2_pi";
+val cos_pi_eq_zero = thm "cos_pi_eq_zero";
+val DERIV_cos_add = thm "DERIV_cos_add";
+val isCont_cos = thm "isCont_cos";
+val isCont_sin = thm "isCont_sin";
+val isCont_exp = thm "isCont_exp";
+val sin_zero_abs_cos_one = thm "sin_zero_abs_cos_one";
+val exp_eq_one_iff = thm "exp_eq_one_iff";
+val cos_one_sin_zero = thm "cos_one_sin_zero";
+val real_root_less_mono = thm "real_root_less_mono";
+val real_root_le_mono = thm "real_root_le_mono";
+val real_root_less_iff = thm "real_root_less_iff";
+val real_root_le_iff = thm "real_root_le_iff";
+val real_root_eq_iff = thm "real_root_eq_iff";
+val real_root_pos_unique = thm "real_root_pos_unique";
+val real_root_mult = thm "real_root_mult";
+val real_root_inverse = thm "real_root_inverse";
+val real_root_divide = thm "real_root_divide";
+val real_sqrt_less_mono = thm "real_sqrt_less_mono";
+val real_sqrt_le_mono = thm "real_sqrt_le_mono";
+val real_sqrt_less_iff = thm "real_sqrt_less_iff";
+val real_sqrt_le_iff = thm "real_sqrt_le_iff";
+val real_sqrt_eq_iff = thm "real_sqrt_eq_iff";
+val real_sqrt_sos_less_one_iff = thm "real_sqrt_sos_less_one_iff";
+val real_sqrt_sos_eq_one_iff = thm "real_sqrt_sos_eq_one_iff";
+val real_divide_square_eq = thm "real_divide_square_eq";
+val real_sqrt_sum_squares_gt_zero1 = thm "real_sqrt_sum_squares_gt_zero1";
+val real_sqrt_sum_squares_gt_zero2 = thm "real_sqrt_sum_squares_gt_zero2";
+val real_sqrt_sum_squares_gt_zero3 = thm "real_sqrt_sum_squares_gt_zero3";
+val real_sqrt_sum_squares_gt_zero3a = thm "real_sqrt_sum_squares_gt_zero3a";
+val real_sqrt_sum_squares_eq_cancel = thm "real_sqrt_sum_squares_eq_cancel";
+val real_sqrt_sum_squares_eq_cancel2 = thm "real_sqrt_sum_squares_eq_cancel2";
+val cos_x_y_ge_minus_one = thm "cos_x_y_ge_minus_one";
+val cos_x_y_ge_minus_one1a = thm "cos_x_y_ge_minus_one1a";
+val cos_x_y_le_one = thm "cos_x_y_le_one";
+val cos_x_y_le_one2 = thm "cos_x_y_le_one2";
+val cos_abs_x_y_ge_minus_one = thm "cos_abs_x_y_ge_minus_one";
+val cos_abs_x_y_le_one = thm "cos_abs_x_y_le_one";
+val minus_pi_less_zero = thm "minus_pi_less_zero";
+val arcos_ge_minus_pi = thm "arcos_ge_minus_pi";
+val sin_x_y_disj = thm "sin_x_y_disj";
+val cos_x_y_disj = thm "cos_x_y_disj";
+val real_sqrt_divide_less_zero = thm "real_sqrt_divide_less_zero";
+val polar_ex1 = thm "polar_ex1";
+val real_sum_squares_cancel2a = thm "real_sum_squares_cancel2a";
+val polar_ex2 = thm "polar_ex2";
+val polar_Ex = thm "polar_Ex";
+val real_sqrt_ge_abs1 = thm "real_sqrt_ge_abs1";
+val real_sqrt_ge_abs2 = thm "real_sqrt_ge_abs2";
+val real_sqrt_two_gt_zero = thm "real_sqrt_two_gt_zero";
+val real_sqrt_two_ge_zero = thm "real_sqrt_two_ge_zero";
+val real_sqrt_two_gt_one = thm "real_sqrt_two_gt_one";
+val STAR_exp_ln = thm "STAR_exp_ln";
+val hypreal_add_Infinitesimal_gt_zero = thm "hypreal_add_Infinitesimal_gt_zero";
+val NSDERIV_exp_ln_one = thm "NSDERIV_exp_ln_one";
+val DERIV_exp_ln_one = thm "DERIV_exp_ln_one";
+val isCont_inv_fun = thm "isCont_inv_fun";
+val isCont_inv_fun_inv = thm "isCont_inv_fun_inv";
+val LIM_fun_gt_zero = thm "LIM_fun_gt_zero";
+val LIM_fun_less_zero = thm "LIM_fun_less_zero";
+val LIM_fun_not_zero = thm "LIM_fun_not_zero";
+*}
   
 end 
diff -r 4b3d280ef06a -r 89840837108e src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Mon Jul 26 15:48:50 2004 +0200
+++ b/src/HOL/IsaMakefile	Mon Jul 26 17:34:52 2004 +0200
@@ -154,8 +154,7 @@
   Hyperreal/Lim.thy Hyperreal/Log.thy\
   Hyperreal/MacLaurin_lemmas.ML Hyperreal/MacLaurin.thy Hyperreal/NatStar.thy\
   Hyperreal/NSA.thy Hyperreal/NthRoot.thy Hyperreal/Poly.thy\
-  Hyperreal/SEQ.ML Hyperreal/SEQ.thy Hyperreal/Series.thy\
-  Hyperreal/Star.thy Hyperreal/Transcendental.ML\
+  Hyperreal/SEQ.ML Hyperreal/SEQ.thy Hyperreal/Series.thy Hyperreal/Star.thy \
   Hyperreal/Transcendental.thy Hyperreal/fuf.ML Hyperreal/hypreal_arith.ML \
   Complex/Complex_Main.thy Complex/CLim.thy Complex/CSeries.thy\
   Complex/CStar.thy Complex/Complex.thy Complex/ComplexBin.thy\
diff -r 4b3d280ef06a -r 89840837108e src/HOL/Real/RealDef.thy
--- a/src/HOL/Real/RealDef.thy	Mon Jul 26 15:48:50 2004 +0200
+++ b/src/HOL/Real/RealDef.thy	Mon Jul 26 17:34:52 2004 +0200
@@ -798,6 +798,7 @@
 
 subsubsection{*Density of the Reals*}
 
+(*????FIXME: rename d1, d2 to x, y*)
 lemma real_lbound_gt_zero:
      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
 apply (rule_tac x = " (min d1 d2) /2" in exI)
diff -r 4b3d280ef06a -r 89840837108e src/HOL/Ring_and_Field.thy
--- a/src/HOL/Ring_and_Field.thy	Mon Jul 26 15:48:50 2004 +0200
+++ b/src/HOL/Ring_and_Field.thy	Mon Jul 26 17:34:52 2004 +0200
@@ -729,6 +729,15 @@
   thus ?thesis  by simp
   qed
 
+lemma inverse_unique: 
+  assumes ab: "a*b = 1"
+  shows "inverse a = (b::'a::field)"
+proof -
+  have "a \<noteq> 0" using ab by auto
+  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
+  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
+qed
+
 lemma nonzero_inverse_mult_distrib: 
       assumes anz: "a \<noteq> 0"
           and bnz: "b \<noteq> 0"