# HG changeset patch
# User paulson
# Date 1079689863 -3600
# Node ID cc61fd03e589c9714c732ec86ef625389a702032
# Parent  758e7acdea2f3f0764ac5dae9b23e73aa501f09d
conversion of Hyperreal/Lim to new-style

diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/HTranscendental.thy
--- a/src/HOL/Hyperreal/HTranscendental.thy	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/HTranscendental.thy	Fri Mar 19 10:51:03 2004 +0100
@@ -573,7 +573,7 @@
 apply (frule STAR_sin_Infinitesimal_divide
                [OF Infinitesimal_pi_divide_HNatInfinite 
                    pi_divide_HNatInfinite_not_zero])
-apply (auto simp add: hypreal_inverse_distrib)
+apply (auto simp add: inverse_mult_distrib)
 apply (rule approx_SReal_mult_cancel [of "inverse (hypreal_of_real pi)"])
 apply (auto intro: SReal_inverse simp add: divide_inverse mult_ac)
 done
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/HyperDef.thy
--- a/src/HOL/Hyperreal/HyperDef.thy	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/HyperDef.thy	Fri Mar 19 10:51:03 2004 +0100
@@ -543,64 +543,17 @@
     by (auto dest: order_le_less_trans simp add: hrabs_def linorder_not_le)
 qed
 
-lemma hypreal_mult_1_right: "z * (1::hypreal) = z"
-  by (rule Ring_and_Field.mult_1_right)
-
-lemma hypreal_mult_minus_1 [simp]: "(- (1::hypreal)) * z = -z"
-by simp
-
-lemma hypreal_mult_minus_1_right [simp]: "z * (- (1::hypreal)) = -z"
-by (subst hypreal_mult_commute, simp)
-
-(*Used ONCE: in NSA.ML*)
-lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
-by (simp add: hypreal_add_commute)
-
-(*Used ONCE: in Lim.ML*)
-lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
-by (auto simp add: hypreal_add_assoc)
-
 lemma hypreal_eq_minus_iff: "((x::hypreal) = y) = (x + - y = 0)"
 apply auto
 apply (rule Ring_and_Field.add_right_cancel [of _ "-y", THEN iffD1], auto)
 done
 
-(*Used 3 TIMES: in Lim.ML*)
-lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
-by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
-
 lemma hypreal_mult_left_cancel: "(c::hypreal) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
 by auto
     
 lemma hypreal_mult_right_cancel: "(c::hypreal) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
 by auto
 
-lemma hypreal_mult_not_0: "[| x \<noteq> 0; y \<noteq> 0 |] ==> x * y \<noteq> (0::hypreal)"
-by simp
-
-lemma hypreal_minus_inverse: "inverse(-x) = -inverse(x::hypreal)"
-  by (rule Ring_and_Field.inverse_minus_eq)
-
-lemma hypreal_inverse_distrib: "inverse(x*y) = inverse(x)*inverse(y::hypreal)"
-  by (rule Ring_and_Field.inverse_mult_distrib)
-
-
-subsection{* Division lemmas *}
-
-lemma hypreal_divide_one: "x/(1::hypreal) = x"
-by (simp add: hypreal_divide_def)
-
-
-(** As with multiplication, pull minus signs OUT of the / operator **)
-
-lemma hypreal_add_divide_distrib: "(x+y)/(z::hypreal) = x/z + y/z"
-  by (rule Ring_and_Field.add_divide_distrib)
-
-lemma hypreal_inverse_add:
-     "[|(x::hypreal) \<noteq> 0;  y \<noteq> 0 |]   
-      ==> inverse(x) + inverse(y) = (x + y)*inverse(x*y)"
-by (simp add: Ring_and_Field.inverse_add mult_assoc)
-
 
 subsection{*The Embedding @{term hypreal_of_real} Preserves Field and 
       Order Properties*}
@@ -832,15 +785,11 @@
 val hypreal_add_zero_right = thm "hypreal_add_zero_right";
 val hypreal_add_minus = thm "hypreal_add_minus";
 val hypreal_add_minus_left = thm "hypreal_add_minus_left";
-val hypreal_minus_distrib1 = thm "hypreal_minus_distrib1";
 val hypreal_mult_congruent2 = thm "hypreal_mult_congruent2";
 val hypreal_mult = thm "hypreal_mult";
 val hypreal_mult_commute = thm "hypreal_mult_commute";
 val hypreal_mult_assoc = thm "hypreal_mult_assoc";
 val hypreal_mult_1 = thm "hypreal_mult_1";
-val hypreal_mult_1_right = thm "hypreal_mult_1_right";
-val hypreal_mult_minus_1 = thm "hypreal_mult_minus_1";
-val hypreal_mult_minus_1_right = thm "hypreal_mult_minus_1_right";
 val hypreal_zero_not_eq_one = thm "hypreal_zero_not_eq_one";
 val hypreal_inverse_congruent = thm "hypreal_inverse_congruent";
 val hypreal_inverse = thm "hypreal_inverse";
@@ -848,14 +797,9 @@
 val hypreal_mult_inverse_left = thm "hypreal_mult_inverse_left";
 val hypreal_mult_left_cancel = thm "hypreal_mult_left_cancel";
 val hypreal_mult_right_cancel = thm "hypreal_mult_right_cancel";
-val hypreal_mult_not_0 = thm "hypreal_mult_not_0";
-val hypreal_minus_inverse = thm "hypreal_minus_inverse";
-val hypreal_inverse_distrib = thm "hypreal_inverse_distrib";
 val hypreal_not_refl2 = thm "hypreal_not_refl2";
 val hypreal_less = thm "hypreal_less";
 val hypreal_eq_minus_iff = thm "hypreal_eq_minus_iff";
-val hypreal_eq_minus_iff3 = thm "hypreal_eq_minus_iff3";
-val hypreal_not_eq_minus_iff = thm "hypreal_not_eq_minus_iff";
 val hypreal_le = thm "hypreal_le";
 val hypreal_le_refl = thm "hypreal_le_refl";
 val hypreal_le_linear = thm "hypreal_le_linear";
@@ -872,9 +816,6 @@
 val hypreal_of_real_zero = thm "hypreal_of_real_zero";
 val hypreal_of_real_inverse = thm "hypreal_of_real_inverse";
 val hypreal_of_real_divide = thm "hypreal_of_real_divide";
-val hypreal_divide_one = thm "hypreal_divide_one";
-val hypreal_add_divide_distrib = thm "hypreal_add_divide_distrib";
-val hypreal_inverse_add = thm "hypreal_inverse_add";
 val hypreal_zero_num = thm "hypreal_zero_num";
 val hypreal_one_num = thm "hypreal_one_num";
 val hypreal_omega_gt_zero = thm "hypreal_omega_gt_zero";
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/Integration.ML
--- a/src/HOL/Hyperreal/Integration.ML	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/Integration.ML	Fri Mar 19 10:51:03 2004 +0100
@@ -508,7 +508,7 @@
 by (Asm_full_simp_tac 2 THEN arith_tac 2);
 by (ALLGOALS (thin_tac "ALL xa. \
 \             xa ~= x & abs (xa + - x) < s --> \
-\             abs ((f xa + - f x) / (xa + - x) + - f' x) * 2 < e"));
+\             abs ((f xa - f x) / (xa - x) + - f' x) * 2 < e"));
 by (dres_inst_tac [("x","v")] spec 1 THEN Auto_tac);
 by (arith_tac 1);
 by (dres_inst_tac [("x","u")] spec 1 THEN Auto_tac);
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/Lim.ML
--- a/src/HOL/Hyperreal/Lim.ML	Fri Mar 19 10:50:06 2004 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2307 +0,0 @@
-(*  Title       : Lim.ML
-    Author      : Jacques D. Fleuriot
-    Copyright   : 1998  University of Cambridge
-    Description : Theory of limits, continuity and
-                  differentiation of real=>real functions
-*)
-
-fun ARITH_PROVE str = prove_goal thy str
-                      (fn prems => [cut_facts_tac prems 1,arith_tac 1]);
-
-
-(*---------------------------------------------------------------
-   Theory of limits, continuity and differentiation of
-   real=>real functions
- ----------------------------------------------------------------*)
-
-Goalw [LIM_def] "(%x. k) -- x --> k";
-by Auto_tac;
-qed "LIM_const";
-Addsimps [LIM_const];
-
-(***-----------------------------------------------------------***)
-(***  Some Purely Standard Proofs - Can be used for comparison ***)
-(***-----------------------------------------------------------***)
-
-(*---------------
-    LIM_add
- ---------------*)
-Goalw [LIM_def]
-     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + g(x)) -- x --> (l + m)";
-by (Clarify_tac 1);
-by (REPEAT(dres_inst_tac [("x","r/2")] spec 1));
-by (Asm_full_simp_tac 1);
-by (Clarify_tac 1);
-by (res_inst_tac [("x","s"),("y","sa")] linorder_cases 1);
-by (res_inst_tac [("x","s")] exI 1);
-by (res_inst_tac [("x","sa")] exI 2);
-by (res_inst_tac [("x","sa")] exI 3);
-by Safe_tac;
-by (REPEAT(dres_inst_tac [("x","xa")] spec 1)
-    THEN step_tac (claset() addSEs [order_less_trans]) 1);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 2)
-    THEN step_tac (claset() addSEs [order_less_trans]) 2);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 3)
-    THEN step_tac (claset() addSEs [order_less_trans]) 3);
-by (ALLGOALS(rtac (abs_sum_triangle_ineq RS order_le_less_trans)));
-by (ALLGOALS(rtac (real_sum_of_halves RS subst)));
-by (auto_tac (claset() addIs [add_strict_mono],simpset()));
-qed "LIM_add";
-
-Goalw [LIM_def] "f -- a --> L ==> (%x. -f(x)) -- a --> -L";
-by (subgoal_tac "ALL x. abs(- f x + L) = abs(f x + - L)" 1);
-by (Asm_full_simp_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [real_abs_def]) 1);
-qed "LIM_minus";
-
-(*----------------------------------------------
-     LIM_add_minus
- ----------------------------------------------*)
-Goal "[| f -- x --> l; g -- x --> m |] \
-\     ==> (%x. f(x) + -g(x)) -- x --> (l + -m)";
-by (blast_tac (claset() addDs [LIM_add,LIM_minus]) 1);
-qed "LIM_add_minus";
-
-(*----------------------------------------------
-     LIM_zero
- ----------------------------------------------*)
-Goal "f -- a --> l ==> (%x. f(x) + -l) -- a --> 0";
-by (res_inst_tac [("a1","l")] ((right_minus RS subst)) 1);
-by (rtac LIM_add_minus 1 THEN Auto_tac);
-qed "LIM_zero";
-
-(*--------------------------
-   Limit not zero
- --------------------------*)
-Goalw [LIM_def] "k \\<noteq> 0 ==> ~ ((%x. k) -- x --> 0)";
-by (res_inst_tac [("x","k"),("y","0")] linorder_cases 1);
-by (auto_tac (claset(), simpset() addsimps [real_abs_def]));
-by (res_inst_tac [("x","-k")] exI 1);
-by (res_inst_tac [("x","k")] exI 2);
-by Auto_tac;
-by (ALLGOALS(dres_inst_tac [("y","s")] real_dense));
-by Safe_tac;
-by (ALLGOALS(res_inst_tac [("x","r + x")] exI));
-by Auto_tac;
-qed "LIM_not_zero";
-
-(* [| k \\<noteq> 0; (%x. k) -- x --> 0 |] ==> R *)
-bind_thm("LIM_not_zeroE", LIM_not_zero RS notE);
-
-Goal "(%x. k) -- x --> L ==> k = L";
-by (rtac ccontr 1);
-by (dtac LIM_zero 1);
-by (rtac LIM_not_zeroE 1 THEN assume_tac 2);
-by (arith_tac 1);
-qed "LIM_const_eq";
-
-(*------------------------
-     Limit is Unique
- ------------------------*)
-Goal "[| f -- x --> L; f -- x --> M |] ==> L = M";
-by (dtac LIM_minus 1);
-by (dtac LIM_add 1 THEN assume_tac 1);
-by (auto_tac (claset() addSDs [LIM_const_eq RS sym],  simpset()));
-qed "LIM_unique";
-
-(*-------------
-    LIM_mult_zero
- -------------*)
-Goalw [LIM_def]
-     "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0";
-by Safe_tac;
-by (dres_inst_tac [("x","1")] spec 1);
-by (dres_inst_tac [("x","r")] spec 1);
-by (cut_facts_tac [real_zero_less_one] 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_mult]) 1);
-by (Clarify_tac 1);
-by (res_inst_tac [("x","s"),("y","sa")]
-    linorder_cases 1);
-by (res_inst_tac [("x","s")] exI 1);
-by (res_inst_tac [("x","sa")] exI 2);
-by (res_inst_tac [("x","sa")] exI 3);
-by Safe_tac;
-by (REPEAT(dres_inst_tac [("x","xa")] spec 1)
-    THEN step_tac (claset() addSEs [order_less_trans]) 1);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 2)
-    THEN step_tac (claset() addSEs [order_less_trans]) 2);
-by (REPEAT(dres_inst_tac [("x","xa")] spec 3)
-    THEN step_tac (claset() addSEs [order_less_trans]) 3);
-by (ALLGOALS(res_inst_tac [("t","r")] (real_mult_1 RS subst)));
-by (ALLGOALS(rtac abs_mult_less));
-by Auto_tac;
-qed "LIM_mult_zero";
-
-Goalw [LIM_def] "(%x. x) -- a --> a";
-by Auto_tac;
-qed "LIM_self";
-
-(*--------------------------------------------------------------
-   Limits are equal for functions equal except at limit point
- --------------------------------------------------------------*)
-Goalw [LIM_def]
-     "[| \\<forall>x. x \\<noteq> a --> (f x = g x) |] \
-\     ==> (f -- a --> l) = (g -- a --> l)";
-by (auto_tac (claset(), simpset() addsimps [real_add_minus_iff]));
-qed "LIM_equal";
-
-Goal "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] \
-\     ==> f -- a --> l";
-by (dtac LIM_add 1 THEN assume_tac 1);
-by (auto_tac (claset(), simpset() addsimps [real_add_assoc]));
-qed "LIM_trans";
-
-(***-------------------------------------------------------------***)
-(***           End of Purely Standard Proofs                     ***)
-(***-------------------------------------------------------------***)
-(*--------------------------------------------------------------
-       Standard and NS definitions of Limit
- --------------------------------------------------------------*)
-Goalw [LIM_def,NSLIM_def,approx_def]
-      "f -- x --> L ==> f -- x --NS> L";
-by (asm_full_simp_tac
-    (simpset() addsimps [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by Safe_tac;
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),
-      simpset() addsimps [real_add_minus_iff, starfun, hypreal_minus,
-                          hypreal_of_real_def, hypreal_add]));
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2 THEN Step_tac 1);
-by (dres_inst_tac [("x","u")] spec 1 THEN Clarify_tac 1);
-by (dres_inst_tac [("x","s")] spec 1 THEN Clarify_tac 1);
-by (subgoal_tac "\\<forall>n::nat. (xa n) \\<noteq> x & \
-\                    abs ((xa n) + - x) < s --> abs (f (xa n) + - L) < u" 1);
-by (Blast_tac 2);
-by (dtac FreeUltrafilterNat_all 1);
-by (Ultra_tac 1);
-qed "LIM_NSLIM";
-
-(*---------------------------------------------------------------------
-    Limit: NS definition ==> standard definition
- ---------------------------------------------------------------------*)
-
-Goal "\\<forall>s. 0 < s --> (\\<exists>xa.  xa \\<noteq> x & \
-\        abs (xa + - x) < s  & r \\<le> abs (f xa + -L)) \
-\     ==> \\<forall>n::nat. \\<exists>xa.  xa \\<noteq> x & \
-\             abs(xa + -x) < inverse(real(Suc n)) & r \\<le> abs(f xa + -L)";
-by (Clarify_tac 1);
-by (cut_inst_tac [("n1","n")]
-    (real_of_nat_Suc_gt_zero RS positive_imp_inverse_positive) 1);
-by Auto_tac;
-qed "lemma_LIM";
-
-Goal "\\<forall>s. 0 < s --> (\\<exists>xa.  xa \\<noteq> x & \
-\        abs (xa + - x) < s  & r \\<le> abs (f xa + -L)) \
-\     ==> \\<exists>X. \\<forall>n::nat. X n \\<noteq> x & \
-\               abs(X n + -x) < inverse(real(Suc n)) & r \\<le> abs(f (X n) + -L)";
-by (dtac lemma_LIM 1);
-by (dtac choice 1);
-by (Blast_tac 1);
-qed "lemma_skolemize_LIM2";
-
-Goal "\\<forall>n. X n \\<noteq> x & \
-\         abs (X n + - x) < inverse (real(Suc n)) & \
-\         r \\<le> abs (f (X n) + - L) ==> \
-\         \\<forall>n. abs (X n + - x) < inverse (real(Suc n))";
-by (Auto_tac );
-qed "lemma_simp";
-
-(*-------------------
-    NSLIM => LIM
- -------------------*)
-
-Goalw [LIM_def,NSLIM_def,approx_def]
-     "f -- x --NS> L ==> f -- x --> L";
-by (asm_full_simp_tac
-    (simpset() addsimps [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by (EVERY1[Step_tac, rtac ccontr, Asm_full_simp_tac]);
-by (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1);
-by (dtac lemma_skolemize_LIM2 1);
-by Safe_tac;
-by (dres_inst_tac [("x","Abs_hypreal(hyprel``{X})")] spec 1);
-by (auto_tac
-    (claset(),
-     simpset() addsimps [starfun, hypreal_minus,
-                         hypreal_of_real_def,hypreal_add]));
-by (dtac (lemma_simp RS real_seq_to_hypreal_Infinitesimal) 1);
-by (asm_full_simp_tac
-    (simpset() addsimps
-       [Infinitesimal_FreeUltrafilterNat_iff,hypreal_of_real_def,
-        hypreal_minus, hypreal_add]) 1);
-by (Blast_tac 1);
-by (dtac spec 1 THEN dtac mp 1 THEN assume_tac 1);
-by (dtac FreeUltrafilterNat_all 1);
-by (Ultra_tac 1);
-qed "NSLIM_LIM";
-
-
-(**** Key result ****)
-Goal "(f -- x --> L) = (f -- x --NS> L)";
-by (blast_tac (claset() addIs [LIM_NSLIM,NSLIM_LIM]) 1);
-qed "LIM_NSLIM_iff";
-
-(*-------------------------------------------------------------------*)
-(*   Proving properties of limits using nonstandard definition and   *)
-(*   hence, the properties hold for standard limits as well          *)
-(*-------------------------------------------------------------------*)
-(*------------------------------------------------
-      NSLIM_mult and hence (trivially) LIM_mult
- ------------------------------------------------*)
-
-Goalw [NSLIM_def]
-     "[| f -- x --NS> l; g -- x --NS> m |] \
-\     ==> (%x. f(x) * g(x)) -- x --NS> (l * m)";
-by (auto_tac (claset() addSIs [approx_mult_HFinite],  simpset()));
-qed "NSLIM_mult";
-
-Goal "[| f -- x --> l; g -- x --> m |] \
-\     ==> (%x. f(x) * g(x)) -- x --> (l * m)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_mult]) 1);
-qed "LIM_mult2";
-
-(*----------------------------------------------
-      NSLIM_add and hence (trivially) LIM_add
-      Note the much shorter proof
- ----------------------------------------------*)
-Goalw [NSLIM_def]
-     "[| f -- x --NS> l; g -- x --NS> m |] \
-\     ==> (%x. f(x) + g(x)) -- x --NS> (l + m)";
-by (auto_tac (claset() addSIs [approx_add], simpset()));
-qed "NSLIM_add";
-
-Goal "[| f -- x --> l; g -- x --> m |] \
-\     ==> (%x. f(x) + g(x)) -- x --> (l + m)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_add]) 1);
-qed "LIM_add2";
-
-(*----------------------------------------------
-     NSLIM_const
- ----------------------------------------------*)
-Goalw [NSLIM_def] "(%x. k) -- x --NS> k";
-by Auto_tac;
-qed "NSLIM_const";
-
-Addsimps [NSLIM_const];
-
-Goal "(%x. k) -- x --> k";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1);
-qed "LIM_const2";
-
-(*----------------------------------------------
-     NSLIM_minus
- ----------------------------------------------*)
-Goalw [NSLIM_def]
-      "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L";
-by Auto_tac;
-qed "NSLIM_minus";
-
-Goal "f -- a --> L ==> (%x. -f(x)) -- a --> -L";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_minus]) 1);
-qed "LIM_minus2";
-
-(*----------------------------------------------
-     NSLIM_add_minus
- ----------------------------------------------*)
-Goal "[| f -- x --NS> l; g -- x --NS> m |] \
-\     ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)";
-by (blast_tac (claset() addDs [NSLIM_add,NSLIM_minus]) 1);
-qed "NSLIM_add_minus";
-
-Goal "[| f -- x --> l; g -- x --> m |] \
-\     ==> (%x. f(x) + -g(x)) -- x --> (l + -m)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff,
-    NSLIM_add_minus]) 1);
-qed "LIM_add_minus2";
-
-(*-----------------------------
-    NSLIM_inverse
- -----------------------------*)
-Goalw [NSLIM_def]
-     "[| f -- a --NS> L;  L \\<noteq> 0 |] \
-\     ==> (%x. inverse(f(x))) -- a --NS> (inverse L)";
-by (Clarify_tac 1);
-by (dtac spec 1);
-by (auto_tac (claset(),
-              simpset() addsimps [hypreal_of_real_approx_inverse]));
-qed "NSLIM_inverse";
-
-Goal "[| f -- a --> L; \
-\        L \\<noteq> 0 |] ==> (%x. inverse(f(x))) -- a --> (inverse L)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_inverse]) 1);
-qed "LIM_inverse";
-
-(*------------------------------
-    NSLIM_zero
- ------------------------------*)
-Goal "f -- a --NS> l ==> (%x. f(x) + -l) -- a --NS> 0";
-by (res_inst_tac [("a1","l")] ((right_minus RS subst)) 1);
-by (rtac NSLIM_add_minus 1 THEN Auto_tac);
-qed "NSLIM_zero";
-
-Goal "f -- a --> l ==> (%x. f(x) + -l) -- a --> 0";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_zero]) 1);
-qed "LIM_zero2";
-
-Goal "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l";
-by (dres_inst_tac [("g","%x. l"),("m","l")] NSLIM_add 1);
-by (auto_tac (claset(),simpset() addsimps [real_diff_def, real_add_assoc]));
-qed "NSLIM_zero_cancel";
-
-Goal "(%x. f(x) - l) -- x --> 0 ==> f -- x --> l";
-by (dres_inst_tac [("g","%x. l"),("m","l")] LIM_add 1);
-by (auto_tac (claset(),simpset() addsimps [real_diff_def, real_add_assoc]));
-qed "LIM_zero_cancel";
-
-
-(*--------------------------
-   NSLIM_not_zero
- --------------------------*)
-Goalw [NSLIM_def] "k \\<noteq> 0 ==> ~ ((%x. k) -- x --NS> 0)";
-by Auto_tac;
-by (res_inst_tac [("x","hypreal_of_real x + epsilon")] exI 1);
-by (auto_tac (claset() addIs [Infinitesimal_add_approx_self RS approx_sym],
-              simpset() addsimps [hypreal_epsilon_not_zero]));
-qed "NSLIM_not_zero";
-
-(* [| k \\<noteq> 0; (%x. k) -- x --NS> 0 |] ==> R *)
-bind_thm("NSLIM_not_zeroE", NSLIM_not_zero RS notE);
-
-Goal "k \\<noteq> 0 ==> ~ ((%x. k) -- x --> 0)";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_not_zero]) 1);
-qed "LIM_not_zero2";
-
-(*-------------------------------------
-   NSLIM of constant function
- -------------------------------------*)
-Goal "(%x. k) -- x --NS> L ==> k = L";
-by (rtac ccontr 1);
-by (dtac NSLIM_zero 1);
-by (rtac NSLIM_not_zeroE 1 THEN assume_tac 2);
-by (arith_tac 1);
-qed "NSLIM_const_eq";
-
-Goal "(%x. k) -- x --> L ==> k = L";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff,
-    NSLIM_const_eq]) 1);
-qed "LIM_const_eq2";
-
-(*------------------------
-     NS Limit is Unique
- ------------------------*)
-(* can actually be proved more easily by unfolding def! *)
-Goal "[| f -- x --NS> L; f -- x --NS> M |] ==> L = M";
-by (dtac NSLIM_minus 1);
-by (dtac NSLIM_add 1 THEN assume_tac 1);
-by (auto_tac (claset() addSDs [NSLIM_const_eq RS sym], simpset()));
-qed "NSLIM_unique";
-
-Goal "[| f -- x --> L; f -- x --> M |] ==> L = M";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_unique]) 1);
-qed "LIM_unique2";
-
-(*--------------------
-    NSLIM_mult_zero
- --------------------*)
-Goal "[| f -- x --NS> 0; g -- x --NS> 0 |] \
-\         ==> (%x. f(x)*g(x)) -- x --NS> 0";
-by (dtac NSLIM_mult 1 THEN Auto_tac);
-qed "NSLIM_mult_zero";
-
-(* we can use the corresponding thm LIM_mult2 *)
-(* for standard definition of limit           *)
-
-Goal "[| f -- x --> 0; g -- x --> 0 |] \
-\     ==> (%x. f(x)*g(x)) -- x --> 0";
-by (dtac LIM_mult2 1 THEN Auto_tac);
-qed "LIM_mult_zero2";
-
-(*----------------------------
-    NSLIM_self
- ----------------------------*)
-Goalw [NSLIM_def] "(%x. x) -- a --NS> a";
-by (auto_tac (claset() addIs [starfun_Idfun_approx],simpset()));
-qed "NSLIM_self";
-
-Goal "(%x. x) -- a --> a";
-by (simp_tac (simpset() addsimps [LIM_NSLIM_iff,NSLIM_self]) 1);
-qed "LIM_self2";
-
-(*-----------------------------------------------------------------------------
-   Derivatives and Continuity - NS and Standard properties
- -----------------------------------------------------------------------------*)
-(*---------------
-    Continuity
- ---------------*)
-
-Goalw [isNSCont_def]
-      "[| isNSCont f a; y \\<approx> hypreal_of_real a |] \
-\           ==> ( *f* f) y \\<approx> hypreal_of_real (f a)";
-by (Blast_tac 1);
-qed "isNSContD";
-
-Goalw [isNSCont_def,NSLIM_def]
-      "isNSCont f a ==> f -- a --NS> (f a) ";
-by (Blast_tac 1);
-qed "isNSCont_NSLIM";
-
-Goalw [isNSCont_def,NSLIM_def]
-      "f -- a --NS> (f a) ==> isNSCont f a";
-by Auto_tac;
-by (res_inst_tac [("Q","y = hypreal_of_real a")]
-    (excluded_middle RS disjE) 1);
-by Auto_tac;
-qed "NSLIM_isNSCont";
-
-(*-----------------------------------------------------
-    NS continuity can be defined using NS Limit in
-    similar fashion to standard def of continuity
- -----------------------------------------------------*)
-Goal "(isNSCont f a) = (f -- a --NS> (f a))";
-by (blast_tac (claset() addIs [isNSCont_NSLIM,NSLIM_isNSCont]) 1);
-qed "isNSCont_NSLIM_iff";
-
-(*----------------------------------------------
-  Hence, NS continuity can be given
-  in terms of standard limit
- ---------------------------------------------*)
-Goal "(isNSCont f a) = (f -- a --> (f a))";
-by (asm_full_simp_tac (simpset() addsimps
-    [LIM_NSLIM_iff,isNSCont_NSLIM_iff]) 1);
-qed "isNSCont_LIM_iff";
-
-(*-----------------------------------------------
-  Moreover, it's trivial now that NS continuity
-  is equivalent to standard continuity
- -----------------------------------------------*)
-Goalw [isCont_def] "(isNSCont f a) = (isCont f a)";
-by (rtac isNSCont_LIM_iff 1);
-qed "isNSCont_isCont_iff";
-
-(*----------------------------------------
-  Standard continuity ==> NS continuity
- ----------------------------------------*)
-Goal "isCont f a ==> isNSCont f a";
-by (etac (isNSCont_isCont_iff RS iffD2) 1);
-qed "isCont_isNSCont";
-
-(*----------------------------------------
-  NS continuity ==> Standard continuity
- ----------------------------------------*)
-Goal "isNSCont f a ==> isCont f a";
-by (etac (isNSCont_isCont_iff RS iffD1) 1);
-qed "isNSCont_isCont";
-
-(*--------------------------------------------------------------------------
-                 Alternative definition of continuity
- --------------------------------------------------------------------------*)
-(* Prove equivalence between NS limits - *)
-(* seems easier than using standard def  *)
-Goalw [NSLIM_def] "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)";
-by Auto_tac;
-by (dres_inst_tac [("x","hypreal_of_real a + x")] spec 1);
-by (dres_inst_tac [("x","-hypreal_of_real a + x")] spec 2);
-by Safe_tac;
-by (Asm_full_simp_tac 1);
-by (rtac ((mem_infmal_iff RS iffD2) RS
-    (Infinitesimal_add_approx_self RS approx_sym)) 1);
-by (rtac (approx_minus_iff2 RS iffD1) 4);
-by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 3);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 2);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 4);
-by (auto_tac (claset(),
-       simpset() addsimps [starfun, hypreal_of_real_def, hypreal_minus,
-              hypreal_add, real_add_assoc, approx_refl, hypreal_zero_def]));
-qed "NSLIM_h_iff";
-
-Goal "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)";
-by (rtac NSLIM_h_iff 1);
-qed "NSLIM_isCont_iff";
-
-Goal "(f -- a --> f a) = ((%h. f(a + h)) -- 0 --> f(a))";
-by (simp_tac (simpset() addsimps [LIM_NSLIM_iff, NSLIM_isCont_iff]) 1);
-qed "LIM_isCont_iff";
-
-Goalw [isCont_def] "(isCont f x) = ((%h. f(x + h)) -- 0 --> f(x))";
-by (simp_tac (simpset() addsimps [LIM_isCont_iff]) 1);
-qed "isCont_iff";
-
-(*--------------------------------------------------------------------------
-   Immediate application of nonstandard criterion for continuity can offer
-   very simple proofs of some standard property of continuous functions
- --------------------------------------------------------------------------*)
-(*------------------------
-     sum continuous
- ------------------------*)
-Goal "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a";
-by (auto_tac (claset() addIs [approx_add],
-              simpset() addsimps [isNSCont_isCont_iff RS sym, isNSCont_def]));
-qed "isCont_add";
-
-(*------------------------
-     mult continuous
- ------------------------*)
-Goal "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a";
-by (auto_tac (claset() addSIs [starfun_mult_HFinite_approx],
-              simpset() delsimps [starfun_mult RS sym]
-			addsimps [isNSCont_isCont_iff RS sym, isNSCont_def]));
-qed "isCont_mult";
-
-(*-------------------------------------------
-     composition of continuous functions
-     Note very short straightforard proof!
- ------------------------------------------*)
-Goal "[| isCont f a; isCont g (f a) |] \
-\     ==> isCont (g o f) a";
-by (auto_tac (claset(),simpset() addsimps [isNSCont_isCont_iff RS sym,
-              isNSCont_def,starfun_o RS sym]));
-qed "isCont_o";
-
-Goal "[| isCont f a; isCont g (f a) |] \
-\     ==> isCont (%x. g (f x)) a";
-by (auto_tac (claset() addDs [isCont_o],simpset() addsimps [o_def]));
-qed "isCont_o2";
-
-Goalw [isNSCont_def] "isNSCont f a ==> isNSCont (%x. - f x) a";
-by Auto_tac;
-qed "isNSCont_minus";
-
-Goal "isCont f a ==> isCont (%x. - f x) a";
-by (auto_tac (claset(),simpset() addsimps [isNSCont_isCont_iff RS sym,
-              isNSCont_minus]));
-qed "isCont_minus";
-
-Goalw [isCont_def]
-      "[| isCont f x; f x \\<noteq> 0 |] ==> isCont (%x. inverse (f x)) x";
-by (blast_tac (claset() addIs [LIM_inverse]) 1);
-qed "isCont_inverse";
-
-Goal "[| isNSCont f x; f x \\<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x";
-by (auto_tac (claset() addIs [isCont_inverse],simpset() addsimps
-    [isNSCont_isCont_iff]));
-qed "isNSCont_inverse";
-
-Goalw [real_diff_def]
-      "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a";
-by (auto_tac (claset() addIs [isCont_add,isCont_minus],simpset()));
-qed "isCont_diff";
-
-Goalw [isCont_def]  "isCont (%x. k) a";
-by (Simp_tac 1);
-qed "isCont_const";
-Addsimps [isCont_const];
-
-Goalw [isNSCont_def]  "isNSCont (%x. k) a";
-by (Simp_tac 1);
-qed "isNSCont_const";
-Addsimps [isNSCont_const];
-
-Goalw [isNSCont_def]  "isNSCont abs a";
-by (auto_tac (claset() addIs [approx_hrabs],
-              simpset() addsimps [hypreal_of_real_hrabs RS sym,
-                                  starfun_rabs_hrabs]));
-qed "isNSCont_rabs";
-Addsimps [isNSCont_rabs];
-
-Goal "isCont abs a";
-by (auto_tac (claset(), simpset() addsimps [isNSCont_isCont_iff RS sym]));
-qed "isCont_rabs";
-Addsimps [isCont_rabs];
-
-(****************************************************************
-(%* Leave as commented until I add topology theory or remove? *%)
-(%*------------------------------------------------------------
-  Elementary topology proof for a characterisation of
-  continuity now: a function f is continuous if and only
-  if the inverse image, {x. f(x) \\<in> A}, of any open set A
-  is always an open set
- ------------------------------------------------------------*%)
-Goal "[| isNSopen A; \\<forall>x. isNSCont f x |] \
-\              ==> isNSopen {x. f x \\<in> A}";
-by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
-by (dtac (mem_monad_approx RS approx_sym) 1);
-by (dres_inst_tac [("x","a")] spec 1);
-by (dtac isNSContD 1 THEN assume_tac 1);
-by (dtac bspec 1 THEN assume_tac 1);
-by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
-by (blast_tac (claset() addIs [starfun_mem_starset]) 1);
-qed "isNSCont_isNSopen";
-
-Goalw [isNSCont_def]
-          "\\<forall>A. isNSopen A --> isNSopen {x. f x \\<in> A} \
-\              ==> isNSCont f x";
-by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
-     (approx_minus_iff RS iffD2)],simpset() addsimps
-      [Infinitesimal_def,SReal_iff]));
-by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
-by (etac (isNSopen_open_interval RSN (2,impE)) 1);
-by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
-by (dres_inst_tac [("x","x")] spec 1);
-by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
-    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
-qed "isNSopen_isNSCont";
-
-Goal "(\\<forall>x. isNSCont f x) = \
-\     (\\<forall>A. isNSopen A --> isNSopen {x. f(x) \\<in> A})";
-by (blast_tac (claset() addIs [isNSCont_isNSopen,
-    isNSopen_isNSCont]) 1);
-qed "isNSCont_isNSopen_iff";
-
-(%*------- Standard version of same theorem --------*%)
-Goal "(\\<forall>x. isCont f x) = \
-\         (\\<forall>A. isopen A --> isopen {x. f(x) \\<in> A})";
-by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
-              simpset() addsimps [isNSopen_isopen_iff RS sym,
-              isNSCont_isCont_iff RS sym]));
-qed "isCont_isopen_iff";
-*******************************************************************)
-
-(*-----------------------------------------------------------------
-                        Uniform continuity
- ------------------------------------------------------------------*)
-Goalw [isNSUCont_def]
-      "[| isNSUCont f; x \\<approx> y|] ==> ( *f* f) x \\<approx> ( *f* f) y";
-by (Blast_tac 1);
-qed "isNSUContD";
-
-Goalw [isUCont_def,isCont_def,LIM_def]
-     "isUCont f ==> isCont f x";
-by (Clarify_tac 1);
-by (dtac spec 1);
-by (Blast_tac 1);
-qed "isUCont_isCont";
-
-Goalw [isNSUCont_def,isUCont_def,approx_def]
-     "isUCont f ==> isNSUCont f";
-by (asm_full_simp_tac (simpset() addsimps
-    [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by Safe_tac;
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
-    hypreal_minus, hypreal_add]));
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2 THEN Step_tac 1);
-by (dres_inst_tac [("x","u")] spec 1 THEN Clarify_tac 1);
-by (dres_inst_tac [("x","s")] spec 1 THEN Clarify_tac 1);
-by (subgoal_tac "\\<forall>n::nat. abs ((xa n) + - (xb n)) < s --> abs (f (xa n) + - f (xb n)) < u" 1);
-by (Blast_tac 2);
-by (thin_tac "\\<forall>x y. abs (x + - y) < s --> abs (f x + - f y) < u" 1);
-by (dtac FreeUltrafilterNat_all 1);
-by (Ultra_tac 1);
-qed "isUCont_isNSUCont";
-
-Goal "\\<forall>s. 0 < s --> (\\<exists>z y. abs (z + - y) < s & r \\<le> abs (f z + -f y)) \
-\     ==> \\<forall>n::nat. \\<exists>z y.  \
-\              abs(z + -y) < inverse(real(Suc n)) & \
-\              r \\<le> abs(f z + -f y)";
-by (Clarify_tac 1);
-by (cut_inst_tac [("n1","n")]
-    (real_of_nat_Suc_gt_zero RS positive_imp_inverse_positive) 1);
-by Auto_tac;
-qed "lemma_LIMu";
-
-Goal "\\<forall>s. 0 < s --> (\\<exists>z y. abs (z + - y) < s  & r \\<le> abs (f z + -f y)) \
-\     ==> \\<exists>X Y. \\<forall>n::nat. \
-\              abs(X n + -(Y n)) < inverse(real(Suc n)) & \
-\              r \\<le> abs(f (X n) + -f (Y n))";
-by (dtac lemma_LIMu 1);
-by (dtac choice 1);
-by Safe_tac;
-by (dtac choice 1);
-by (Blast_tac 1);
-qed "lemma_skolemize_LIM2u";
-
-Goal "\\<forall>n. abs (X n + -Y n) < inverse (real(Suc n)) & \
-\         r \\<le> abs (f (X n) + - f(Y n)) ==> \
-\         \\<forall>n. abs (X n + - Y n) < inverse (real(Suc n))";
-by (Auto_tac );
-qed "lemma_simpu";
-
-Goalw [isNSUCont_def,isUCont_def,approx_def]
-     "isNSUCont f ==> isUCont f";
-by (asm_full_simp_tac (simpset() addsimps
-                       [Infinitesimal_FreeUltrafilterNat_iff]) 1);
-by (EVERY1[Step_tac, rtac ccontr, Asm_full_simp_tac]);
-by (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1);
-by (dtac lemma_skolemize_LIM2u 1);
-by Safe_tac;
-by (dres_inst_tac [("x","Abs_hypreal(hyprel``{X})")] spec 1);
-by (dres_inst_tac [("x","Abs_hypreal(hyprel``{Y})")] spec 1);
-by (asm_full_simp_tac
-    (simpset() addsimps [starfun, hypreal_minus,hypreal_add]) 1);
-by Auto_tac;
-by (dtac (lemma_simpu RS real_seq_to_hypreal_Infinitesimal2) 1);
-by (asm_full_simp_tac (simpset() addsimps
-     [Infinitesimal_FreeUltrafilterNat_iff, hypreal_minus,hypreal_add]) 1);
-by (Blast_tac 1);
-by (rotate_tac 2 1);
-by (dres_inst_tac [("x","r")] spec 1);
-by (Clarify_tac 1);
-by (dtac FreeUltrafilterNat_all 1);
-by (Ultra_tac 1);
-qed "isNSUCont_isUCont";
-
-(*------------------------------------------------------------------
-                         Derivatives
- ------------------------------------------------------------------*)
-Goalw [deriv_def]
-      "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --> D)";
-by (Blast_tac 1);
-qed "DERIV_iff";
-
-Goalw [deriv_def]
-      "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --NS> D)";
-by (simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1);
-qed "DERIV_NS_iff";
-
-Goalw [deriv_def]
-      "DERIV f x :> D \
-\      ==> (%h. (f(x + h) + - f(x))/h) -- 0 --> D";
-by (Blast_tac 1);
-qed "DERIVD";
-
-Goalw [deriv_def] "DERIV f x :> D ==> \
-\          (%h. (f(x + h) + - f(x))/h) -- 0 --NS> D";
-by (asm_full_simp_tac (simpset() addsimps [LIM_NSLIM_iff]) 1);
-qed "NS_DERIVD";
-
-(* Uniqueness *)
-Goalw [deriv_def]
-      "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E";
-by (blast_tac (claset() addIs [LIM_unique]) 1);
-qed "DERIV_unique";
-
-Goalw [nsderiv_def]
-     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E";
-by (cut_facts_tac [Infinitesimal_epsilon, hypreal_epsilon_not_zero] 1);
-by (auto_tac (claset() addSDs [inst "x" "epsilon" bspec]
-                       addSIs [inj_hypreal_of_real RS injD]
-                       addDs [approx_trans3],
-              simpset()));
-qed "NSDeriv_unique";
-
-(*------------------------------------------------------------------------
-                          Differentiable
- ------------------------------------------------------------------------*)
-
-Goalw [differentiable_def]
-      "f differentiable x ==> \\<exists>D. DERIV f x :> D";
-by (assume_tac 1);
-qed "differentiableD";
-
-Goalw [differentiable_def]
-      "DERIV f x :> D ==> f differentiable x";
-by (Blast_tac 1);
-qed "differentiableI";
-
-Goalw [NSdifferentiable_def]
-      "f NSdifferentiable x ==> \\<exists>D. NSDERIV f x :> D";
-by (assume_tac 1);
-qed "NSdifferentiableD";
-
-Goalw [NSdifferentiable_def]
-      "NSDERIV f x :> D ==> f NSdifferentiable x";
-by (Blast_tac 1);
-qed "NSdifferentiableI";
-
-(*--------------------------------------------------------
-      Alternative definition for differentiability
- -------------------------------------------------------*)
-
-Goalw [LIM_def]
- "((%h. (f(a + h) + - f(a))/h) -- 0 --> D) = \
-\ ((%x. (f(x) + -f(a)) / (x + -a)) -- a --> D)";
-by Safe_tac;
-by (ALLGOALS(dtac spec));
-by Safe_tac;
-by (Blast_tac 1 THEN Blast_tac 2);
-by (ALLGOALS(res_inst_tac [("x","s")] exI));
-by Safe_tac;
-by (dres_inst_tac [("x","x + -a")] spec 1);
-by (dres_inst_tac [("x","x + a")] spec 2);
-by (auto_tac (claset(), simpset() addsimps add_ac));
-qed "DERIV_LIM_iff";
-
-Goalw [deriv_def] "(DERIV f x :> D) = \
-\         ((%z. (f(z) + -f(x)) / (z + -x)) -- x --> D)";
-by (simp_tac (simpset() addsimps [DERIV_LIM_iff]) 1);
-qed "DERIV_iff2";
-
-(*--------------------------------------------------------
-  Equivalence of NS and standard defs of differentiation
- -------------------------------------------------------*)
-(*-------------------------------------------
-   First NSDERIV in terms of NSLIM
- -------------------------------------------*)
-
-(*--- first equivalence ---*)
-Goalw [nsderiv_def,NSLIM_def]
-      "(NSDERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --NS> D)";
-by Auto_tac;
-by (dres_inst_tac [("x","xa")] bspec 1);
-by (rtac ccontr 3);
-by (dres_inst_tac [("x","h")] spec 3);
-by (auto_tac (claset(),
-              simpset() addsimps [mem_infmal_iff, starfun_lambda_cancel]));
-qed "NSDERIV_NSLIM_iff";
-
-(*--- second equivalence ---*)
-Goal "(NSDERIV f x :> D) = \
-\         ((%z. (f(z) + -f(x)) / (z + -x)) -- x --NS> D)";
-by (full_simp_tac (simpset() addsimps
-     [NSDERIV_NSLIM_iff, DERIV_LIM_iff, LIM_NSLIM_iff RS sym]) 1);
-qed "NSDERIV_NSLIM_iff2";
-
-(* while we're at it! *)
-Goalw [real_diff_def]
-     "(NSDERIV f x :> D) = \
-\     (\\<forall>xa. \
-\       xa \\<noteq> hypreal_of_real x & xa \\<approx> hypreal_of_real x --> \
-\       ( *f* (%z. (f z - f x) / (z - x))) xa \\<approx> hypreal_of_real D)";
-by (auto_tac (claset(), simpset() addsimps [NSDERIV_NSLIM_iff2, NSLIM_def]));
-qed "NSDERIV_iff2";
-
-
-Goal "(NSDERIV f x :> D) ==> \
-\    (\\<forall>u. \
-\       u \\<approx> hypreal_of_real x --> \
-\       ( *f* (%z. f z - f x)) u \\<approx> hypreal_of_real D * (u - hypreal_of_real x))";
-by (auto_tac (claset(), simpset() addsimps [NSDERIV_iff2]));
-by (case_tac "u = hypreal_of_real x" 1);
-by (auto_tac (claset(), simpset() addsimps [hypreal_diff_def]));
-by (dres_inst_tac [("x","u")] spec 1);
-by Auto_tac;
-by (dres_inst_tac [("c","u - hypreal_of_real x"),("b","hypreal_of_real D")]
-     approx_mult1 1);
-by (ALLGOALS(dtac (hypreal_not_eq_minus_iff RS iffD1)));
-by (subgoal_tac "( *f* (%z. z - x)) u \\<noteq> (0::hypreal)" 2);
-by (auto_tac (claset(),
-    simpset() addsimps [real_diff_def, hypreal_diff_def,
-		(approx_minus_iff RS iffD1) RS (mem_infmal_iff RS iffD2),
-			Infinitesimal_subset_HFinite RS subsetD]));
-qed "NSDERIVD5";
-
-Goal "(NSDERIV f x :> D) ==> \
-\     (\\<forall>h \\<in> Infinitesimal. \
-\              (( *f* f)(hypreal_of_real x + h) - \
-\                hypreal_of_real (f x))\\<approx> (hypreal_of_real D) * h)";
-by (auto_tac (claset(),simpset() addsimps [nsderiv_def]));
-by (case_tac "h = (0::hypreal)" 1);
-by (auto_tac (claset(),simpset() addsimps [hypreal_diff_def]));
-by (dres_inst_tac [("x","h")] bspec 1);
-by (dres_inst_tac [("c","h")] approx_mult1 2);
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD],
-              simpset() addsimps [hypreal_diff_def]));
-qed "NSDERIVD4";
-
-Goal "(NSDERIV f x :> D) ==> \
-\     (\\<forall>h \\<in> Infinitesimal - {0}. \
-\              (( *f* f)(hypreal_of_real x + h) - \
-\                hypreal_of_real (f x))\\<approx> (hypreal_of_real D) * h)";
-by (auto_tac (claset(),simpset() addsimps [nsderiv_def]));
-by (rtac ccontr 1 THEN dres_inst_tac [("x","h")] bspec 1);
-by (dres_inst_tac [("c","h")] approx_mult1 2);
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite RS subsetD],
-              simpset() addsimps [hypreal_mult_assoc, hypreal_diff_def]));
-qed "NSDERIVD3";
-
-(*--------------------------------------------------------------
-          Now equivalence between NSDERIV and DERIV
- -------------------------------------------------------------*)
-Goalw [deriv_def] "(NSDERIV f x :> D) = (DERIV f x :> D)";
-by (simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff,LIM_NSLIM_iff]) 1);
-qed "NSDERIV_DERIV_iff";
-
-(*---------------------------------------------------
-         Differentiability implies continuity
-         nice and simple "algebraic" proof
- --------------------------------------------------*)
-Goalw [nsderiv_def]
-      "NSDERIV f x :> D ==> isNSCont f x";
-by (auto_tac (claset(),simpset() addsimps
-        [isNSCont_NSLIM_iff,NSLIM_def]));
-by (dtac (approx_minus_iff RS iffD1) 1);
-by (dtac (hypreal_not_eq_minus_iff RS iffD1) 1);
-by (dres_inst_tac [("x","-hypreal_of_real x + xa")] bspec 1);
-by (asm_full_simp_tac (simpset() addsimps
-    [hypreal_add_assoc RS sym]) 2);
-by (auto_tac (claset(),simpset() addsimps
-    [mem_infmal_iff RS sym,hypreal_add_commute]));
-by (dres_inst_tac [("c","xa + -hypreal_of_real x")] approx_mult1 1);
-by (auto_tac (claset() addIs [Infinitesimal_subset_HFinite
-    RS subsetD],simpset() addsimps [hypreal_mult_assoc]));
-by (dres_inst_tac [("x3","D")] (HFinite_hypreal_of_real RSN
-    (2,Infinitesimal_HFinite_mult) RS (mem_infmal_iff RS iffD1)) 1);
-by (blast_tac (claset() addIs [approx_trans,
-    hypreal_mult_commute RS subst,
-    (approx_minus_iff RS iffD2)]) 1);
-qed "NSDERIV_isNSCont";
-
-(* Now Sandard proof *)
-Goal "DERIV f x :> D ==> isCont f x";
-by (asm_full_simp_tac (simpset() addsimps
-    [NSDERIV_DERIV_iff RS sym, isNSCont_isCont_iff RS sym,
-     NSDERIV_isNSCont]) 1);
-qed "DERIV_isCont";
-
-(*----------------------------------------------------------------------------
-      Differentiation rules for combinations of functions
-      follow from clear, straightforard, algebraic
-      manipulations
- ----------------------------------------------------------------------------*)
-(*-------------------------
-    Constant function
- ------------------------*)
-
-(* use simple constant nslimit theorem *)
-Goal "(NSDERIV (%x. k) x :> 0)";
-by (simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff]) 1);
-qed "NSDERIV_const";
-Addsimps [NSDERIV_const];
-
-Goal "(DERIV (%x. k) x :> 0)";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_const";
-Addsimps [DERIV_const];
-
-(*-----------------------------------------------------
-    Sum of functions- proved easily
- ----------------------------------------------------*)
-
-
-Goal "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |] \
-\     ==> NSDERIV (%x. f x + g x) x :> Da + Db";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff,
-           NSLIM_def]) 1 THEN REPEAT (Step_tac 1));
-by (auto_tac (claset(),
-       simpset() addsimps [add_divide_distrib]));
-by (dres_inst_tac [("b","hypreal_of_real Da"),
-                   ("d","hypreal_of_real Db")] approx_add 1);
-by (auto_tac (claset(), simpset() addsimps add_ac));
-qed "NSDERIV_add";
-
-(* Standard theorem *)
-Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \
-\     ==> DERIV (%x. f x + g x) x :> Da + Db";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_add,
-                                     NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_add";
-
-(*-----------------------------------------------------
-  Product of functions - Proof is trivial but tedious
-  and long due to rearrangement of terms
- ----------------------------------------------------*)
-
-Goal "((a::hypreal)*b) + -(c*d) = (b*(a + -c)) + (c*(b + -d))";
-by (simp_tac (simpset() addsimps [right_distrib]) 1);
-qed "lemma_nsderiv1";
-
-Goal "[| (x + y) / z = hypreal_of_real D + yb; z \\<noteq> 0; \
-\        z \\<in> Infinitesimal; yb \\<in> Infinitesimal |] \
-\     ==> x + y \\<approx> 0";
-by (forw_inst_tac [("c1","z")] (hypreal_mult_right_cancel RS iffD2) 1
-    THEN assume_tac 1);
-by (thin_tac "(x + y) / z = hypreal_of_real D + yb" 1);
-by (auto_tac (claset() addSIs [Infinitesimal_HFinite_mult2, HFinite_add],
-              simpset() addsimps [hypreal_mult_assoc, mem_infmal_iff RS sym]));
-by (etac (Infinitesimal_subset_HFinite RS subsetD) 1);
-qed "lemma_nsderiv2";
-
-
-Goal "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \
-\     ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff, NSLIM_def]) 1);
-by (REPEAT (Step_tac 1));
-by (auto_tac (claset(),
-       simpset() addsimps [starfun_lambda_cancel, lemma_nsderiv1]));
-by (simp_tac (simpset() addsimps [add_divide_distrib]) 1);
-by (REPEAT(dtac (bex_Infinitesimal_iff2 RS iffD2) 1));
-by (auto_tac (claset(),
-        simpset() delsimps [times_divide_eq_right]
-		  addsimps [times_divide_eq_right RS sym]));
-by (dres_inst_tac [("D","Db")] lemma_nsderiv2 1);
-by (dtac (approx_minus_iff RS iffD2 RS (bex_Infinitesimal_iff2 RS iffD2)) 4);
-by (auto_tac (claset() addSIs [approx_add_mono1],
-      simpset() addsimps [left_distrib, right_distrib,
-			  hypreal_mult_commute, hypreal_add_assoc]));
-by (res_inst_tac [("w1","hypreal_of_real Db * hypreal_of_real (f x)")]
-    (hypreal_add_commute RS subst) 1);
-by (auto_tac (claset() addSIs [Infinitesimal_add_approx_self2 RS approx_sym,
-			       Infinitesimal_add, Infinitesimal_mult,
-			       Infinitesimal_hypreal_of_real_mult,
-			       Infinitesimal_hypreal_of_real_mult2],
-	      simpset() addsimps [hypreal_add_assoc RS sym]));
-qed "NSDERIV_mult";
-
-Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \
-\     ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_mult,
-                                           NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_mult";
-
-(*----------------------------
-   Multiplying by a constant
- ---------------------------*)
-Goal "NSDERIV f x :> D \
-\     ==> NSDERIV (%x. c * f x) x :> c*D";
-by (asm_full_simp_tac
-    (HOL_ss addsimps [times_divide_eq_right RS sym, NSDERIV_NSLIM_iff,
-                      minus_mult_right, right_distrib RS sym]) 1);
-by (etac (NSLIM_const RS NSLIM_mult) 1);
-qed "NSDERIV_cmult";
-
-(* let's do the standard proof though theorem *)
-(* LIM_mult2 follows from a NS proof          *)
-
-Goalw [deriv_def]
-      "DERIV f x :> D \
-\      ==> DERIV (%x. c * f x) x :> c*D";
-by (asm_full_simp_tac
-    (HOL_ss addsimps [times_divide_eq_right RS sym, NSDERIV_NSLIM_iff,
-                      minus_mult_right, right_distrib RS sym]) 1);
-by (etac (LIM_const RS LIM_mult2) 1);
-qed "DERIV_cmult";
-
-(*--------------------------------
-   Negation of function
- -------------------------------*)
-Goal "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff]) 1);
-by (dtac NSLIM_minus 1);
-by (subgoal_tac "ALL a::real. ALL b. - a + b = - (a + - b)" 1);
-by (asm_full_simp_tac (HOL_ss addsimps [thm"minus_divide_left" RS sym]) 1);
-by (Asm_full_simp_tac 1);
-qed "NSDERIV_minus";
-
-Goal "DERIV f x :> D \
-\     ==> DERIV (%x. -(f x)) x :> -D";
-by (asm_full_simp_tac (simpset() addsimps
-    [NSDERIV_minus,NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_minus";
-
-(*-------------------------------
-   Subtraction
- ------------------------------*)
-Goal "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \
-\     ==> NSDERIV (%x. f x + -g x) x :> Da + -Db";
-by (blast_tac (claset() addDs [NSDERIV_add,NSDERIV_minus]) 1);
-qed "NSDERIV_add_minus";
-
-Goal "[| DERIV f x :> Da; DERIV g x :> Db |] \
-\     ==> DERIV (%x. f x + -g x) x :> Da + -Db";
-by (blast_tac (claset() addDs [DERIV_add,DERIV_minus]) 1);
-qed "DERIV_add_minus";
-
-Goalw [real_diff_def]
-     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] \
-\     ==> NSDERIV (%x. f x - g x) x :> Da - Db";
-by (blast_tac (claset() addIs [NSDERIV_add_minus]) 1);
-qed "NSDERIV_diff";
-
-Goalw [real_diff_def]
-     "[| DERIV f x :> Da; DERIV g x :> Db |] \
-\      ==> DERIV (%x. f x - g x) x :> Da - Db";
-by (blast_tac (claset() addIs [DERIV_add_minus]) 1);
-qed "DERIV_diff";
-
-(*---------------------------------------------------------------
-                     (NS) Increment
- ---------------------------------------------------------------*)
-Goalw [increment_def]
-      "f NSdifferentiable x ==> \
-\     increment f x h = ( *f* f) (hypreal_of_real(x) + h) + \
-\     -hypreal_of_real (f x)";
-by (Blast_tac 1);
-qed "incrementI";
-
-Goal "NSDERIV f x :> D ==> \
-\    increment f x h = ( *f* f) (hypreal_of_real(x) + h) + \
-\    -hypreal_of_real (f x)";
-by (etac (NSdifferentiableI RS incrementI) 1);
-qed "incrementI2";
-
-(* The Increment theorem -- Keisler p. 65 *)
-Goal "[| NSDERIV f x :> D; h \\<in> Infinitesimal; h \\<noteq> 0 |] \
-\     ==> \\<exists>e \\<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h";
-by (forw_inst_tac [("h","h")] incrementI2 1 THEN rewtac nsderiv_def);
-by (dtac bspec 1 THEN Auto_tac);
-by (dtac (bex_Infinitesimal_iff2 RS iffD2) 1 THEN Step_tac 1);
-by (forw_inst_tac [("b1","hypreal_of_real(D) + y")]
-    ((hypreal_mult_right_cancel RS iffD2)) 1);
-by (thin_tac "(( *f* f) (hypreal_of_real(x) + h) + \
-\   - hypreal_of_real (f x)) / h = hypreal_of_real(D) + y" 2);
-by (assume_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [times_divide_eq_right RS sym]
-             delsimps [times_divide_eq_right]) 1);
-by (auto_tac (claset(),
-              simpset() addsimps [left_distrib]));
-qed "increment_thm";
-
-Goal "[| NSDERIV f x :> D; h \\<approx> 0; h \\<noteq> 0 |] \
-\     ==> \\<exists>e \\<in> Infinitesimal. increment f x h = \
-\             hypreal_of_real(D)*h + e*h";
-by (blast_tac (claset() addSDs [mem_infmal_iff RS iffD2]
-                        addSIs [increment_thm]) 1);
-qed "increment_thm2";
-
-Goal "[| NSDERIV f x :> D; h \\<approx> 0; h \\<noteq> 0 |] \
-\     ==> increment f x h \\<approx> 0";
-by (dtac increment_thm2 1 THEN auto_tac (claset() addSIs
-    [Infinitesimal_HFinite_mult2,HFinite_add],simpset() addsimps
-    [left_distrib RS sym,mem_infmal_iff RS sym]));
-by (etac (Infinitesimal_subset_HFinite RS subsetD) 1);
-qed "increment_approx_zero";
-
-(*---------------------------------------------------------------
-   Similarly to the above, the chain rule admits an entirely
-   straightforward derivation. Compare this with Harrison's
-   HOL proof of the chain rule, which proved to be trickier and
-   required an alternative characterisation of differentiability-
-   the so-called Carathedory derivative. Our main problem is
-   manipulation of terms.
- --------------------------------------------------------------*)
-
-(* lemmas *)
-Goalw [nsderiv_def]
-      "[| NSDERIV g x :> D; \
-\              ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);\
-\              xa \\<in> Infinitesimal;\
-\              xa \\<noteq> 0 \
-\           |] ==> D = 0";
-by (dtac bspec 1);
-by Auto_tac;
-qed "NSDERIV_zero";
-
-(* can be proved differently using NSLIM_isCont_iff *)
-Goalw [nsderiv_def]
-     "[| NSDERIV f x :> D;  h \\<in> Infinitesimal;  h \\<noteq> 0 |]  \
-\     ==> ( *f* f) (hypreal_of_real(x) + h) + -hypreal_of_real(f x) \\<approx> 0";
-by (asm_full_simp_tac (simpset() addsimps
-    [mem_infmal_iff RS sym]) 1);
-by (rtac Infinitesimal_ratio 1);
-by (rtac approx_hypreal_of_real_HFinite 3);
-by Auto_tac;
-qed "NSDERIV_approx";
-
-(*---------------------------------------------------------------
-   from one version of differentiability
-
-                f(x) - f(a)
-              --------------- \\<approx> Db
-                  x - a
- ---------------------------------------------------------------*)
-Goal "[| NSDERIV f (g x) :> Da; \
-\        ( *f* g) (hypreal_of_real(x) + xa) \\<noteq> hypreal_of_real (g x); \
-\        ( *f* g) (hypreal_of_real(x) + xa) \\<approx> hypreal_of_real (g x) \
-\     |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa)) \
-\                  + - hypreal_of_real (f (g x))) \
-\             / (( *f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real (g x)) \
-\            \\<approx> hypreal_of_real(Da)";
-by (auto_tac (claset(),
-       simpset() addsimps [NSDERIV_NSLIM_iff2, NSLIM_def]));
-qed "NSDERIVD1";
-
-(*--------------------------------------------------------------
-   from other version of differentiability
-
-                f(x + h) - f(x)
-               ----------------- \\<approx> Db
-                       h
- --------------------------------------------------------------*)
-Goal "[| NSDERIV g x :> Db; xa \\<in> Infinitesimal; xa \\<noteq> 0 |] \
-\     ==> (( *f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real(g x)) / xa \
-\         \\<approx> hypreal_of_real(Db)";
-by (auto_tac (claset(),
-    simpset() addsimps [NSDERIV_NSLIM_iff, NSLIM_def,
-		        mem_infmal_iff, starfun_lambda_cancel]));
-qed "NSDERIVD2";
-
-Goal "(z::hypreal) \\<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)";
-by Auto_tac;
-qed "lemma_chain";
-
-(*------------------------------------------------------
-  This proof uses both definitions of differentiability.
- ------------------------------------------------------*)
-Goal "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |] \
-\     ==> NSDERIV (f o g) x :> Da * Db";
-by (asm_simp_tac (simpset() addsimps [NSDERIV_NSLIM_iff,
-    NSLIM_def,mem_infmal_iff RS sym]) 1 THEN Step_tac 1);
-by (forw_inst_tac [("f","g")] NSDERIV_approx 1);
-by (auto_tac (claset(),
-              simpset() addsimps [starfun_lambda_cancel2, starfun_o RS sym]));
-by (case_tac "( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real (g x)" 1);
-by (dres_inst_tac [("g","g")] NSDERIV_zero 1);
-by (auto_tac (claset(), simpset() addsimps [hypreal_divide_def]));
-by (res_inst_tac [("z1","( *f* g) (hypreal_of_real(x) + xa) + -hypreal_of_real (g x)"),
-    ("y1","inverse xa")] (lemma_chain RS ssubst) 1);
-by (etac (hypreal_not_eq_minus_iff RS iffD1) 1);
-by (rtac approx_mult_hypreal_of_real 1);
-by (fold_tac [hypreal_divide_def]);
-by (blast_tac (claset() addIs [NSDERIVD1,
-    approx_minus_iff RS iffD2]) 1);
-by (blast_tac (claset() addIs [NSDERIVD2]) 1);
-qed "NSDERIV_chain";
-
-(* standard version *)
-Goal "[| DERIV f (g x) :> Da; \
-\                 DERIV g x :> Db \
-\              |] ==> DERIV (f o g) x :> Da * Db";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym,
-    NSDERIV_chain]) 1);
-qed "DERIV_chain";
-
-Goal "[| DERIV f (g x) :> Da; DERIV g x :> Db |] \
-\     ==> DERIV (%x. f (g x)) x :> Da * Db";
-by (auto_tac (claset() addDs [DERIV_chain], simpset() addsimps [o_def]));
-qed "DERIV_chain2";
-
-(*------------------------------------------------------------------
-           Differentiation of natural number powers
- ------------------------------------------------------------------*)
-Goal "NSDERIV (%x. x) x :> 1";
-by (auto_tac (claset(),
-     simpset() addsimps [NSDERIV_NSLIM_iff, NSLIM_def ,starfun_Id]));
-qed "NSDERIV_Id";
-Addsimps [NSDERIV_Id];
-
-(*derivative of the identity function*)
-Goal "DERIV (%x. x) x :> 1";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff RS sym]) 1);
-qed "DERIV_Id";
-Addsimps [DERIV_Id];
-
-bind_thm ("isCont_Id", DERIV_Id RS DERIV_isCont);
-
-(*derivative of linear multiplication*)
-Goal "DERIV (op * c) x :> c";
-by (cut_inst_tac [("c","c"),("x","x")] (DERIV_Id RS DERIV_cmult) 1);
-by (Asm_full_simp_tac 1);
-qed "DERIV_cmult_Id";
-Addsimps [DERIV_cmult_Id];
-
-Goal "NSDERIV (op * c) x :> c";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff]) 1);
-qed "NSDERIV_cmult_Id";
-Addsimps [NSDERIV_cmult_Id];
-
-Goal "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))";
-by (induct_tac "n" 1);
-by (dtac (DERIV_Id RS DERIV_mult) 2);
-by (auto_tac (claset(),
-              simpset() addsimps [real_of_nat_Suc, left_distrib]));
-by (case_tac "0 < n" 1);
-by (dres_inst_tac [("x","x")] realpow_minus_mult 1);
-by (auto_tac (claset(),
-              simpset() addsimps [real_mult_assoc, real_add_commute]));
-qed "DERIV_pow";
-
-(* NS version *)
-Goal "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))";
-by (simp_tac (simpset() addsimps [NSDERIV_DERIV_iff, DERIV_pow]) 1);
-qed "NSDERIV_pow";
-
-(*---------------------------------------------------------------
-                    Power of -1
- ---------------------------------------------------------------*)
-
-(*Can't get rid of x \\<noteq> 0 because it isn't continuous at zero*)
-Goalw [nsderiv_def]
-     "x \\<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))";
-by (rtac ballI 1 THEN Asm_full_simp_tac 1 THEN Step_tac 1);
-by (ftac Infinitesimal_add_not_zero 1);
-by (asm_full_simp_tac (simpset() addsimps [hypreal_add_commute]) 2);
-by (auto_tac (claset(),
-     simpset() addsimps [starfun_inverse_inverse, realpow_two]
-               delsimps [minus_mult_left RS sym,
-                         minus_mult_right RS sym]));
-by (asm_full_simp_tac
-     (simpset() addsimps [hypreal_inverse_add,
-          hypreal_inverse_distrib RS sym, hypreal_minus_inverse RS sym]
-          @ add_ac @ mult_ac
-       delsimps [inverse_mult_distrib,inverse_minus_eq,
-		 minus_mult_left RS sym,
-                 minus_mult_right RS sym] ) 1);
-by (asm_simp_tac (simpset() addsimps [hypreal_mult_assoc RS sym,
-                                      right_distrib]
-         delsimps [minus_mult_left RS sym,
-                   minus_mult_right RS sym]) 1);
-by (res_inst_tac [("y"," inverse(- hypreal_of_real x * hypreal_of_real x)")]
-                 approx_trans 1);
-by (rtac inverse_add_Infinitesimal_approx2 1);
-by (auto_tac (claset() addSDs [hypreal_of_real_HFinite_diff_Infinitesimal],
-         simpset() addsimps [hypreal_minus_inverse RS sym,
-                             HFinite_minus_iff]));
-by (rtac Infinitesimal_HFinite_mult2 1);
-by Auto_tac;
-qed "NSDERIV_inverse";
-
-
-Goal "x \\<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))";
-by (asm_simp_tac (simpset() addsimps [NSDERIV_inverse,
-         NSDERIV_DERIV_iff RS sym] delsimps [realpow_Suc]) 1);
-qed "DERIV_inverse";
-
-(*--------------------------------------------------------------
-        Derivative of inverse
- -------------------------------------------------------------*)
-Goal "[| DERIV f x :> d; f(x) \\<noteq> 0 |] \
-\     ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))";
-by (rtac (real_mult_commute RS subst) 1);
-by (asm_simp_tac (HOL_ss addsimps [minus_mult_left, power_inverse]) 1);
-by (fold_goals_tac [o_def]);
-by (blast_tac (claset() addSIs [DERIV_chain,DERIV_inverse]) 1);
-qed "DERIV_inverse_fun";
-
-Goal "[| NSDERIV f x :> d; f(x) \\<noteq> 0 |] \
-\     ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff,
-            DERIV_inverse_fun] delsimps [realpow_Suc]) 1);
-qed "NSDERIV_inverse_fun";
-
-(*--------------------------------------------------------------
-        Derivative of quotient
- -------------------------------------------------------------*)
-Goal "[| DERIV f x :> d; DERIV g x :> e; g(x) \\<noteq> 0 |] \
-\      ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) + -(e*f(x))) / (g(x) ^ Suc (Suc 0))";
-by (dres_inst_tac [("f","g")] DERIV_inverse_fun 1);
-by (dtac DERIV_mult 2);
-by (REPEAT(assume_tac 1));
-by (asm_full_simp_tac
-    (simpset() addsimps [real_divide_def, right_distrib,
-                         power_inverse,minus_mult_left] @ mult_ac
-       delsimps [realpow_Suc, minus_mult_right RS sym, minus_mult_left RS sym]) 1);
-qed "DERIV_quotient";
-
-Goal "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \\<noteq> 0 |] \
-\      ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x) \
-\                           + -(e*f(x))) / (g(x) ^ Suc (Suc 0))";
-by (asm_full_simp_tac (simpset() addsimps [NSDERIV_DERIV_iff,
-            DERIV_quotient] delsimps [realpow_Suc]) 1);
-qed "NSDERIV_quotient";
-
-(* ------------------------------------------------------------------------ *)
-(* Caratheodory formulation of derivative at a point: standard proof        *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "(DERIV f x :> l) = \
-\     (\\<exists>g. (\\<forall>z. f z - f x = g z * (z - x)) & isCont g x & g x = l)";
-by Safe_tac;
-by (res_inst_tac
-    [("x","%z. if  z = x then l else (f(z) - f(x)) / (z - x)")] exI 1);
-by (auto_tac (claset(),simpset() addsimps [real_mult_assoc,
-    ARITH_PROVE "z \\<noteq> x ==> z - x \\<noteq> (0::real)"]));
-by (auto_tac (claset(),simpset() addsimps [isCont_iff,DERIV_iff]));
-by (ALLGOALS(rtac (LIM_equal RS iffD1)));
-by (auto_tac (claset(),simpset() addsimps [real_diff_def,real_mult_assoc]));
-qed "CARAT_DERIV";
-
-Goal "NSDERIV f x :> l ==> \
-\     \\<exists>g. (\\<forall>z. f z - f x = g z * (z - x)) & isNSCont g x & g x = l";
-by (auto_tac (claset(),simpset() addsimps [NSDERIV_DERIV_iff,
-    isNSCont_isCont_iff,CARAT_DERIV]));
-qed "CARAT_NSDERIV";
-
-(* How about a NS proof? *)
-Goal "(\\<forall>z. f z - f x = g z * (z - x)) & isNSCont g x & g x = l \
-\     ==> NSDERIV f x :> l";
-by (auto_tac (claset(),
-              simpset() delsimprocs field_cancel_factor
-                        addsimps [NSDERIV_iff2]));
-by (auto_tac (claset(),
-              simpset() addsimps [hypreal_mult_assoc]));
-by (asm_full_simp_tac (simpset() addsimps [hypreal_eq_minus_iff3 RS sym,
-                                           hypreal_diff_def]) 1);
-by (asm_full_simp_tac (simpset() addsimps [isNSCont_def]) 1);
-qed "CARAT_DERIVD";
-
-
-
-(*--------------------------------------------------------------------------*)
-(* Lemmas about nested intervals and proof by bisection (cf.Harrison)       *)
-(* All considerably tidied by lcp                                           *)
-(*--------------------------------------------------------------------------*)
-
-Goal "(\\<forall>n. (f::nat=>real) n \\<le> f (Suc n)) --> f m \\<le> f(m + no)";
-by (induct_tac "no" 1);
-by (auto_tac (claset() addIs [order_trans], simpset()));
-qed_spec_mp "lemma_f_mono_add";
-
-Goal "[| \\<forall>n. f(n) \\<le> f(Suc n); \
-\        \\<forall>n. g(Suc n) \\<le> g(n); \
-\        \\<forall>n. f(n) \\<le> g(n) |] \
-\     ==> Bseq f";
-by (res_inst_tac [("k","f 0"),("K","g 0")] BseqI2 1 THEN rtac allI 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset() addIs [order_trans], simpset()));
-by (res_inst_tac [("y","g(Suc na)")] order_trans 1);
-by (induct_tac "na" 2);
-by (auto_tac (claset() addIs [order_trans], simpset()));
-qed "f_inc_g_dec_Beq_f";
-
-Goal "[| \\<forall>n. f(n) \\<le> f(Suc n); \
-\        \\<forall>n. g(Suc n) \\<le> g(n); \
-\        \\<forall>n. f(n) \\<le> g(n) |] \
-\     ==> Bseq g";
-by (stac (Bseq_minus_iff RS sym) 1);
-by (res_inst_tac [("g","%x. -(f x)")] f_inc_g_dec_Beq_f 1);
-by Auto_tac;
-qed "f_inc_g_dec_Beq_g";
-
-Goal "[| \\<forall>n. f n \\<le> f (Suc n);  convergent f |] ==> f n \\<le> lim f";
-by (rtac (linorder_not_less RS iffD1) 1);
-by (auto_tac (claset(),
-      simpset() addsimps [convergent_LIMSEQ_iff, LIMSEQ_iff, monoseq_Suc]));
-by (dtac real_less_sum_gt_zero 1);
-by (dres_inst_tac [("x","f n + - lim f")] spec 1);
-by Safe_tac;
-by (dres_inst_tac [("P","%na. no\\<le>na --> ?Q na"),("x","no + n")] spec 1);
-by Auto_tac;
-by (subgoal_tac "lim f \\<le> f(no + n)" 1);
-by (induct_tac "no" 2);
-by (auto_tac (claset() addIs [order_trans],
-              simpset() addsimps [real_diff_def, real_abs_def]));
-by (dres_inst_tac [("x","f(no + n)"),("no1","no")]
-    (lemma_f_mono_add RSN (2,order_less_le_trans)) 1);
-by (auto_tac (claset(), simpset() addsimps [add_commute]));
-qed "f_inc_imp_le_lim";
-
-Goal "convergent g ==> lim (%x. - g x) = - (lim g)";
-by (rtac (LIMSEQ_minus RS limI) 1);
-by (asm_full_simp_tac (simpset() addsimps [convergent_LIMSEQ_iff]) 1);
-qed "lim_uminus";
-
-Goal "[| \\<forall>n. g(Suc n) \\<le> g(n);  convergent g |] ==> lim g \\<le> g n";
-by (subgoal_tac "- (g n) \\<le> - (lim g)" 1);
-by (cut_inst_tac [("f", "%x. - (g x)")] f_inc_imp_le_lim 2);
-by (auto_tac (claset(),
-              simpset() addsimps [lim_uminus, convergent_minus_iff RS sym]));
-qed "g_dec_imp_lim_le";
-
-Goal "[| \\<forall>n. f(n) \\<le> f(Suc n); \
-\        \\<forall>n. g(Suc n) \\<le> g(n); \
-\        \\<forall>n. f(n) \\<le> g(n) |] \
-\     ==> \\<exists>l m. l \\<le> m &  ((\\<forall>n. f(n) \\<le> l) & f ----> l) & \
-\                           ((\\<forall>n. m \\<le> g(n)) & g ----> m)";
-by (subgoal_tac "monoseq f & monoseq g" 1);
-by (force_tac (claset(), simpset() addsimps [LIMSEQ_iff,monoseq_Suc]) 2);
-by (subgoal_tac "Bseq f & Bseq g" 1);
-by (blast_tac (claset() addIs [f_inc_g_dec_Beq_f, f_inc_g_dec_Beq_g]) 2);
-by (auto_tac (claset() addSDs [Bseq_monoseq_convergent],
-              simpset() addsimps [convergent_LIMSEQ_iff]));
-by (res_inst_tac [("x","lim f")] exI 1);
-by (res_inst_tac [("x","lim g")] exI 1);
-by (auto_tac (claset() addIs [LIMSEQ_le], simpset()));
-by (auto_tac (claset(),
-              simpset() addsimps [f_inc_imp_le_lim, g_dec_imp_lim_le,
-                                  convergent_LIMSEQ_iff]));
-qed "lemma_nest";
-
-Goal "[| \\<forall>n. f(n) \\<le> f(Suc n); \
-\        \\<forall>n. g(Suc n) \\<le> g(n); \
-\        \\<forall>n. f(n) \\<le> g(n); \
-\        (%n. f(n) - g(n)) ----> 0 |] \
-\     ==> \\<exists>l. ((\\<forall>n. f(n) \\<le> l) & f ----> l) & \
-\               ((\\<forall>n. l \\<le> g(n)) & g ----> l)";
-by (dtac lemma_nest 1 THEN Auto_tac);
-by (subgoal_tac "l = m" 1);
-by (dres_inst_tac [("X","f")] LIMSEQ_diff 2);
-by (auto_tac (claset() addIs [LIMSEQ_unique], simpset()));
-qed "lemma_nest_unique";
-
-
-Goal "a \\<le> b ==> \
-\  \\<forall>n. fst (Bolzano_bisect P a b n) \\<le> snd (Bolzano_bisect P a b n)";
-by (rtac allI 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset(), simpset() addsimps [Let_def, split_def]));
-qed "Bolzano_bisect_le";
-
-Goal "a \\<le> b ==> \
-\  \\<forall>n. fst(Bolzano_bisect P a b n) \\<le> fst (Bolzano_bisect P a b (Suc n))";
-by (rtac allI 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset(),
-              simpset() addsimps [Bolzano_bisect_le, Let_def, split_def]));
-qed "Bolzano_bisect_fst_le_Suc";
-
-Goal "a \\<le> b ==> \
-\  \\<forall>n. snd(Bolzano_bisect P a b (Suc n)) \\<le> snd (Bolzano_bisect P a b n)";
-by (rtac allI 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset(),
-              simpset() addsimps [Bolzano_bisect_le, Let_def, split_def]));
-qed "Bolzano_bisect_Suc_le_snd";
-
-Goal "((x::real) = y / (2 * z)) = (2 * x = y/z)";
-by Auto_tac;
-by (dres_inst_tac [("f","%u. (1/2)*u")] arg_cong 1);
-by Auto_tac;
-qed "eq_divide_2_times_iff";
-
-Goal "a \\<le> b ==> \
-\     snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) = \
-\     (b-a) / (2 ^ n)";
-by (induct_tac "n" 1);
-by (auto_tac (claset(),
-      simpset() addsimps [eq_divide_2_times_iff, add_divide_distrib,
-                          Let_def, split_def]));
-by (auto_tac (claset(),
-              simpset() addsimps (add_ac@[Bolzano_bisect_le, real_diff_def])));
-qed "Bolzano_bisect_diff";
-
-val Bolzano_nest_unique =
-    [Bolzano_bisect_fst_le_Suc, Bolzano_bisect_Suc_le_snd, Bolzano_bisect_le]
-    MRS lemma_nest_unique;
-
-(*P_prem is a looping simprule, so it works better if it isn't an assumption*)
-val P_prem::notP_prem::rest =
-Goal "[| !!a b c. [| P(a,b); P(b,c); a \\<le> b; b \\<le> c|] ==> P(a,c); \
-\        ~ P(a,b);  a \\<le> b |] ==> \
-\     ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))";
-by (cut_facts_tac rest 1);
-by (induct_tac "n" 1);
-by (auto_tac (claset(),
-              simpset() delsimps [surjective_pairing RS sym]
-			addsimps [notP_prem, Let_def, split_def]));
-by (swap_res_tac [P_prem] 1);
-by (assume_tac 1);
-by (auto_tac (claset(), simpset() addsimps [Bolzano_bisect_le]));
-qed "not_P_Bolzano_bisect";
-
-(*Now we re-package P_prem as a formula*)
-Goal "[| \\<forall>a b c. P(a,b) & P(b,c) & a \\<le> b & b \\<le> c --> P(a,c); \
-\        ~ P(a,b);  a \\<le> b |] ==> \
-\     \\<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))";
-by (blast_tac (claset() addSEs [not_P_Bolzano_bisect RSN (2,rev_notE)]) 1);
-qed "not_P_Bolzano_bisect'";
-
-
-Goal "[| \\<forall>a b c. P(a,b) & P(b,c) & a \\<le> b & b \\<le> c --> P(a,c); \
-\        \\<forall>x. \\<exists>d::real. 0 < d & \
-\               (\\<forall>a b. a \\<le> x & x \\<le> b & (b - a) < d --> P(a,b)); \
-\        a \\<le> b |]  \
-\     ==> P(a,b)";
-by (rtac (inst "P1" "P" Bolzano_nest_unique RS exE) 1);
-by (REPEAT (assume_tac 1));
-by (rtac LIMSEQ_minus_cancel 1);
-by (asm_simp_tac (simpset() addsimps [Bolzano_bisect_diff,
-                                      LIMSEQ_divide_realpow_zero]) 1);
-by (rtac ccontr 1);
-by (dtac not_P_Bolzano_bisect' 1);
-by (REPEAT (assume_tac 1));
-by (rename_tac "l" 1);
-by (dres_inst_tac [("x","l")] spec 1 THEN Clarify_tac 1);
-by (rewtac LIMSEQ_def);
-by (dres_inst_tac [("P", "%r. 0<r --> ?Q r"), ("x","d/2")] spec 1);
-by (dres_inst_tac [("P", "%r. 0<r --> ?Q r"), ("x","d/2")] spec 1);
-by (dtac real_less_half_sum 1);
-by Safe_tac;
-(*linear arithmetic bug if we just use Asm_simp_tac*)
-by (ALLGOALS Asm_full_simp_tac);
-by (dres_inst_tac [("x","fst(Bolzano_bisect P a b (no + noa))")] spec 1);
-by (dres_inst_tac [("x","snd(Bolzano_bisect P a b (no + noa))")] spec 1);
-by Safe_tac;
-by (ALLGOALS Asm_simp_tac);
-by (res_inst_tac [("y","abs(fst(Bolzano_bisect P a b(no + noa)) - l) + \
-\                       abs(snd(Bolzano_bisect P a b(no + noa)) - l)")]
-    order_le_less_trans 1);
-by (asm_simp_tac (simpset() addsimps [real_abs_def]) 1);
-by (rtac (real_sum_of_halves RS subst) 1);
-by (rtac add_strict_mono 1);
-by (ALLGOALS
-    (asm_full_simp_tac (simpset() addsimps [symmetric real_diff_def])));
-qed "lemma_BOLZANO";
-
-
-Goal "((\\<forall>a b c. (a \\<le> b & b \\<le> c & P(a,b) & P(b,c)) --> P(a,c)) & \
-\      (\\<forall>x. \\<exists>d::real. 0 < d & \
-\               (\\<forall>a b. a \\<le> x & x \\<le> b & (b - a) < d --> P(a,b)))) \
-\     --> (\\<forall>a b. a \\<le> b --> P(a,b))";
-by (Clarify_tac 1);
-by (blast_tac (claset() addIs [lemma_BOLZANO]) 1);
-qed "lemma_BOLZANO2";
-
-
-(*----------------------------------------------------------------------------*)
-(* Intermediate Value Theorem (prove contrapositive by bisection)             *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| f(a) \\<le> y & y \\<le> f(b); \
-\        a \\<le> b; \
-\        (\\<forall>x. a \\<le> x & x \\<le> b --> isCont f x) |] \
-\     ==> \\<exists>x. a \\<le> x & x \\<le> b & f(x) = y";
-by (rtac contrapos_pp 1);
-by (assume_tac 1);
-by (cut_inst_tac
-    [("P","%(u,v). a \\<le> u & u \\<le> v & v \\<le> b --> ~(f(u) \\<le> y & y \\<le> f(v))")]
-    lemma_BOLZANO2 1);
-by Safe_tac;
-by (ALLGOALS(Asm_full_simp_tac));
-by (asm_full_simp_tac (simpset() addsimps [isCont_iff,LIM_def]) 1);
-by (rtac ccontr 1);
-by (subgoal_tac "a \\<le> x & x \\<le> b" 1);
-by (Asm_full_simp_tac 2);
-by (dres_inst_tac [("P", "%d. 0<d --> ?P d"),("x","1")] spec 2);
-by (Step_tac 2);
-by (Asm_full_simp_tac 2);
-by (Asm_full_simp_tac 2);
-by (REPEAT(blast_tac (claset() addIs [order_trans]) 2));
-by (REPEAT(dres_inst_tac [("x","x")] spec 1));
-by (Asm_full_simp_tac 1);
-by (dres_inst_tac [("P", "%r. ?P r --> (\\<exists>s. 0<s & ?Q r s)"),
-                   ("x","abs(y - f x)")] spec 1);
-by Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps []) 1);
-by (dres_inst_tac [("x","s")] spec 1);
-by (Clarify_tac 1);
-by (cut_inst_tac [("x","f x"),("y","y")] linorder_less_linear 1);
-by Safe_tac;
-by (dres_inst_tac [("x","ba - x")] spec 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [thm"abs_if"])));
-by (dres_inst_tac [("x","aa - x")] spec 1);
-by (case_tac "x \\<le> aa" 1);
-by (ALLGOALS Asm_full_simp_tac);
-by (dres_inst_tac [("z","x"),("w","aa")] real_le_anti_sym 1);
-by (assume_tac 1 THEN Asm_full_simp_tac 1);
-qed "IVT";
-
-
-Goal "[| f(b) \\<le> y & y \\<le> f(a); \
-\        a \\<le> b; \
-\        (\\<forall>x. a \\<le> x & x \\<le> b --> isCont f x) \
-\     |] ==> \\<exists>x. a \\<le> x & x \\<le> b & f(x) = y";
-by (subgoal_tac "- f a \\<le> -y & -y \\<le> - f b" 1);
-by (thin_tac "f b \\<le> y & y \\<le> f a" 1);
-by (dres_inst_tac [("f","%x. - f x")] IVT 1);
-by (auto_tac (claset() addIs [isCont_minus],simpset()));
-qed "IVT2";
-
-
-(*HOL style here: object-level formulations*)
-Goal "(f(a) \\<le> y & y \\<le> f(b) & a \\<le> b & \
-\     (\\<forall>x. a \\<le> x & x \\<le> b --> isCont f x)) \
-\     --> (\\<exists>x. a \\<le> x & x \\<le> b & f(x) = y)";
-by (blast_tac (claset() addIs [IVT]) 1);
-qed "IVT_objl";
-
-Goal "(f(b) \\<le> y & y \\<le> f(a) & a \\<le> b & \
-\     (\\<forall>x. a \\<le> x & x \\<le> b --> isCont f x)) \
-\     --> (\\<exists>x. a \\<le> x & x \\<le> b & f(x) = y)";
-by (blast_tac (claset() addIs [IVT2]) 1);
-qed "IVT2_objl";
-
-(*---------------------------------------------------------------------------*)
-(* By bisection, function continuous on closed interval is bounded above     *)
-(*---------------------------------------------------------------------------*)
-
-Goal "abs (real x) = real (x::nat)";
-by (auto_tac (claset() addIs [abs_eqI1], simpset()));
-qed "abs_real_of_nat_cancel";
-Addsimps [abs_real_of_nat_cancel];
-
-Goal "~ abs(x) + (1::real) < x";
-by (asm_full_simp_tac (simpset() addsimps [linorder_not_less]) 1);
-by (auto_tac (claset() addIs [abs_ge_self RS order_trans],simpset()));
-qed "abs_add_one_not_less_self";
-Addsimps [abs_add_one_not_less_self];
-
-
-Goal "[| a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |]\
-\     ==> \\<exists>M. \\<forall>x. a \\<le> x & x \\<le> b --> f(x) \\<le> M";
-by (cut_inst_tac [("P","%(u,v). a \\<le> u & u \\<le> v & v \\<le> b --> \
-\                         (\\<exists>M. \\<forall>x. u \\<le> x & x \\<le> v --> f x \\<le> M)")]
-    lemma_BOLZANO2 1);
-by Safe_tac;
-by (ALLGOALS Asm_full_simp_tac);
-by (rename_tac "x xa ya M Ma" 1);
-by (cut_inst_tac [("x","M"),("y","Ma")] linorder_linear 1);
-by Safe_tac;
-by (res_inst_tac [("x","Ma")] exI 1);
-by (Clarify_tac 1);
-by (cut_inst_tac [("x","xb"),("y","xa")] linorder_linear 1);
-by (Force_tac 1);
-by (res_inst_tac [("x","M")] exI 1);
-by (Clarify_tac 1);
-by (cut_inst_tac [("x","xb"),("y","xa")] linorder_linear 1);
-by (Force_tac 1);
-by (case_tac "a \\<le> x & x \\<le> b" 1);
-by (res_inst_tac [("x","1")] exI 2);
-by (Force_tac 2);
-by (asm_full_simp_tac (simpset() addsimps [LIM_def,isCont_iff]) 1);
-by (dres_inst_tac [("x","x")] spec 1 THEN Auto_tac);
-by (thin_tac "\\<forall>M. \\<exists>x. a \\<le> x & x \\<le> b & ~ f x \\<le> M" 1);
-by (dres_inst_tac [("x","1")] spec 1);
-by Auto_tac;
-by (res_inst_tac [("x","s")] exI 1 THEN Clarify_tac 1);
-by (res_inst_tac [("x","abs(f x) + 1")] exI 1 THEN Clarify_tac 1);
-by (dres_inst_tac [("x","xa - x")] spec 1);
-by (auto_tac (claset(), simpset() addsimps [abs_ge_self]));
-by (REPEAT (arith_tac 1));
-qed "isCont_bounded";
-
-(*----------------------------------------------------------------------------*)
-(* Refine the above to existence of least upper bound                         *)
-(*----------------------------------------------------------------------------*)
-
-Goal "((\\<exists>x. x \\<in> S) & (\\<exists>y. isUb UNIV S (y::real))) --> \
-\     (\\<exists>t. isLub UNIV S t)";
-by (blast_tac (claset() addIs [reals_complete]) 1);
-qed "lemma_reals_complete";
-
-Goal "[| a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |] \
-\        ==> \\<exists>M. (\\<forall>x. a \\<le> x & x \\<le> b --> f(x) \\<le> M) & \
-\                  (\\<forall>N. N < M --> (\\<exists>x. a \\<le> x & x \\<le> b & N < f(x)))";
-by (cut_inst_tac [("S","Collect (%y. \\<exists>x. a \\<le> x & x \\<le> b & y = f x)")]
-    lemma_reals_complete 1);
-by Auto_tac;
-by (dtac isCont_bounded 1 THEN assume_tac 1);
-by (auto_tac (claset(),simpset() addsimps [isUb_def,leastP_def,
-    isLub_def,setge_def,setle_def]));
-by (rtac exI 1 THEN Auto_tac);
-by (REPEAT(dtac spec 1) THEN Auto_tac);
-by (dres_inst_tac [("x","x")] spec 1);
-by (auto_tac (claset() addSIs [(linorder_not_less RS iffD1)],simpset()));
-qed "isCont_has_Ub";
-
-(*----------------------------------------------------------------------------*)
-(* Now show that it attains its upper bound                                   *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |] \
-\        ==> \\<exists>M. (\\<forall>x. a \\<le> x & x \\<le> b --> f(x) \\<le> M) & \
-\                  (\\<exists>x. a \\<le> x & x \\<le> b & f(x) = M)";
-by (ftac isCont_has_Ub 1 THEN assume_tac 1);
-by (Clarify_tac 1);
-by (res_inst_tac [("x","M")] exI 1);
-by (Asm_full_simp_tac 1);
-by (rtac ccontr 1);
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> f x < M" 1 THEN Step_tac 1);
-by (rtac ccontr 2 THEN dtac (linorder_not_less RS iffD1) 2);
-by (dres_inst_tac [("z","M")] real_le_anti_sym 2);
-by (REPEAT(Blast_tac 2));
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> isCont (%x. inverse(M - f x)) x" 1);
-by Safe_tac;
-by (EVERY[rtac isCont_inverse 2, rtac isCont_diff 2, rtac notI 4]);
-by (ALLGOALS(asm_full_simp_tac (simpset() addsimps [diff_eq_eq])));
-by (Blast_tac 2);
-by (subgoal_tac
-    "\\<exists>k. \\<forall>x. a \\<le> x & x \\<le> b --> (%x. inverse(M - (f x))) x \\<le> k" 1);
-by (rtac isCont_bounded 2);
-by Safe_tac;
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> 0 < inverse(M - f(x))" 1);
-by (Asm_full_simp_tac 1);
-by Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps [less_diff_eq]) 2);
-by (subgoal_tac
-    "\\<forall>x. a \\<le> x & x \\<le> b --> (%x. inverse(M - (f x))) x < (k + 1)" 1);
-by Safe_tac;
-by (res_inst_tac [("y","k")] order_le_less_trans 2);
-by (asm_full_simp_tac (simpset() addsimps [zero_less_one]) 3);
-by (Asm_full_simp_tac 2);
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> \
-\                inverse(k + 1) < inverse((%x. inverse(M - (f x))) x)" 1);
-by Safe_tac;
-by (rtac less_imp_inverse_less 2);
-by (ALLGOALS Asm_full_simp_tac);
-by (dres_inst_tac [("P", "%N. N<M --> ?Q N"),
-                   ("x","M - inverse(k + 1)")] spec 1);
-by (Step_tac 1 THEN dtac (linorder_not_less RS iffD1) 1);
-by (dtac (le_diff_eq RS iffD1) 1);
-by (REPEAT(dres_inst_tac [("x","a")] spec 1));
-by (Asm_full_simp_tac 1);
-by (asm_full_simp_tac
-    (simpset() addsimps [inverse_eq_divide, pos_divide_le_eq]) 1);
-by (cut_inst_tac [("a","k"),("b","M-f a")] zero_less_mult_iff 1);
-by (Asm_full_simp_tac 1);
-(*last one*)
-by (REPEAT(dres_inst_tac [("x","x")] spec 1));
-by (Asm_full_simp_tac 1);
-qed "isCont_eq_Ub";
-
-
-(*----------------------------------------------------------------------------*)
-(* Same theorem for lower bound                                               *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |] \
-\        ==> \\<exists>M. (\\<forall>x. a \\<le> x & x \\<le> b --> M \\<le> f(x)) & \
-\                  (\\<exists>x. a \\<le> x & x \\<le> b & f(x) = M)";
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> isCont (%x. -(f x)) x" 1);
-by (blast_tac (claset() addIs [isCont_minus]) 2);
-by (dres_inst_tac [("f","(%x. -(f x))")] isCont_eq_Ub 1);
-by Safe_tac;
-by Auto_tac;
-qed "isCont_eq_Lb";
-
-
-(* ------------------------------------------------------------------------- *)
-(* Another version.                                                          *)
-(* ------------------------------------------------------------------------- *)
-
-Goal "[|a \\<le> b; \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x |] \
-\     ==> \\<exists>L M. (\\<forall>x. a \\<le> x & x \\<le> b --> L \\<le> f(x) & f(x) \\<le> M) & \
-\         (\\<forall>y. L \\<le> y & y \\<le> M --> (\\<exists>x. a \\<le> x & x \\<le> b & (f(x) = y)))";
-by (ftac isCont_eq_Lb 1);
-by (ftac isCont_eq_Ub 2);
-by (REPEAT(assume_tac 1));
-by Safe_tac;
-by (res_inst_tac [("x","f x")] exI 1);
-by (res_inst_tac [("x","f xa")] exI 1);
-by (Asm_full_simp_tac 1);
-by Safe_tac;
-by (cut_inst_tac [("x","x"),("y","xa")] linorder_linear 1);
-by Safe_tac;
-by (cut_inst_tac [("f","f"),("a","x"),("b","xa"),("y","y")] IVT_objl 1);
-by (cut_inst_tac [("f","f"),("a","xa"),("b","x"),("y","y")] IVT2_objl 2);
-by Safe_tac;
-by (res_inst_tac [("x","xb")] exI 2);
-by (res_inst_tac [("x","xb")] exI 4);
-by (ALLGOALS(Asm_full_simp_tac));
-qed "isCont_Lb_Ub";
-
-(*----------------------------------------------------------------------------*)
-(* If f'(x) > 0 then x is locally strictly increasing at the right            *)
-(*----------------------------------------------------------------------------*)
-
-Goalw [deriv_def,LIM_def]
-    "[| DERIV f x :> l;  0 < l |] \
-\    ==> \\<exists>d. 0 < d & (\\<forall>h. 0 < h & h < d --> f(x) < f(x + h))";
-by (dtac spec 1 THEN Auto_tac);
-by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac);
-by (subgoal_tac "0 < l*h" 1);
-by (asm_full_simp_tac (simpset() addsimps [zero_less_mult_iff]) 2);
-by (dres_inst_tac [("x","h")] spec 1);
-by (asm_full_simp_tac
-    (simpset() addsimps [real_abs_def, inverse_eq_divide,
-                 pos_le_divide_eq, pos_less_divide_eq]
-              addsplits [split_if_asm]) 1);
-qed "DERIV_left_inc";
-
-val prems = goalw (the_context()) [deriv_def,LIM_def]
-    "[| DERIV f x :> l;  l < 0 |] ==> \
-\      \\<exists>d. 0 < d & (\\<forall>h. 0 < h & h < d --> f(x) < f(x - h))";
-by (cut_facts_tac prems 1);  (*needed because arith removes the assumption l<0*)
-by (dres_inst_tac [("x","-l")] spec 1 THEN Auto_tac);
-by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac);
-by (dres_inst_tac [("x","-h")] spec 1);
-by (asm_full_simp_tac
-    (simpset() addsimps [real_abs_def, inverse_eq_divide,
-                         pos_less_divide_eq,
-                         symmetric real_diff_def]
-               addsplits [split_if_asm]) 1);
-by (subgoal_tac "0 < (f (x - h) - f x)/h" 1);
-by (asm_full_simp_tac (simpset() addsimps [pos_less_divide_eq]) 1);
-by (cut_facts_tac prems 1);
-by (arith_tac 1);
-qed "DERIV_left_dec";
-
-(*????previous proof, revealing arith problem:
-by (dres_inst_tac [("x","-l")] spec 1 THEN Auto_tac);
-by (res_inst_tac [("x","s")] exI 1 THEN Auto_tac);
-by (subgoal_tac "l*h < 0" 1);
-by (asm_full_simp_tac (simpset() addsimps [mult_less_0_iff]) 2);
-by (dres_inst_tac [("x","-h")] spec 1);
-by (asm_full_simp_tac
-    (simpset() addsimps [real_abs_def, inverse_eq_divide,
-                         pos_less_divide_eq,
-                         symmetric real_diff_def]
-               addsplits [split_if_asm]
-               delsimprocs [fast_real_arith_simproc]) 1);
-by (subgoal_tac "0 < (f (x - h) - f x)/h" 1);
-by (arith_tac 2);
-by (asm_full_simp_tac
-    (simpset() addsimps [pos_less_divide_eq]) 1);
-qed "DERIV_left_dec";
-*)
-
-
-Goal "[| DERIV f x :> l; \
-\        \\<exists>d. 0 < d & (\\<forall>y. abs(x - y) < d --> f(y) \\<le> f(x)) |] \
-\     ==> l = 0";
-by (res_inst_tac [("x","l"),("y","0")] linorder_cases 1);
-by Safe_tac;
-by (dtac DERIV_left_dec 1);
-by (dtac DERIV_left_inc 3);
-by Safe_tac;
-by (dres_inst_tac [("d1.0","d"),("d2.0","da")] real_lbound_gt_zero 1);
-by (dres_inst_tac [("d1.0","d"),("d2.0","da")] real_lbound_gt_zero 3);
-by Safe_tac;
-by (dres_inst_tac [("x","x - e")] spec 1);
-by (dres_inst_tac [("x","x + e")] spec 2);
-by (auto_tac (claset(), simpset() addsimps [real_abs_def]));
-qed "DERIV_local_max";
-
-(*----------------------------------------------------------------------------*)
-(* Similar theorem for a local minimum                                        *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| DERIV f x :> l; \
-\        \\<exists>d::real. 0 < d & (\\<forall>y. abs(x - y) < d --> f(x) \\<le> f(y)) |] \
-\     ==> l = 0";
-by (dtac (DERIV_minus RS DERIV_local_max) 1);
-by Auto_tac;
-qed "DERIV_local_min";
-
-(*----------------------------------------------------------------------------*)
-(* In particular if a function is locally flat                                *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| DERIV f x :> l; \
-\        \\<exists>d. 0 < d & (\\<forall>y. abs(x - y) < d --> f(x) = f(y)) |] \
-\     ==> l = 0";
-by (auto_tac (claset() addSDs [DERIV_local_max],simpset()));
-qed "DERIV_local_const";
-
-(*----------------------------------------------------------------------------*)
-(* Lemma about introducing open ball in open interval                         *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a < x;  x < b |] ==> \
-\       \\<exists>d::real. 0 < d &  (\\<forall>y. abs(x - y) < d --> a < y & y < b)";
-by (simp_tac (simpset() addsimps [abs_interval_iff]) 1);
-by (cut_inst_tac [("x","x - a"),("y","b - x")] linorder_linear 1);
-by Safe_tac;
-by (res_inst_tac [("x","x - a")] exI 1);
-by (res_inst_tac [("x","b - x")] exI 2);
-by Auto_tac;
-by (auto_tac (claset(),simpset() addsimps [less_diff_eq]));
-qed "lemma_interval_lt";
-
-Goal "[| a < x;  x < b |] ==> \
-\       \\<exists>d::real. 0 < d &  (\\<forall>y. abs(x - y) < d --> a \\<le> y & y \\<le> b)";
-by (dtac lemma_interval_lt 1);
-by Auto_tac;
-by (auto_tac (claset() addSIs [exI] ,simpset()));
-qed "lemma_interval";
-
-(*-----------------------------------------------------------------------
-            Rolle's Theorem
-   If f is defined and continuous on the finite closed interval [a,b]
-   and differentiable a least on the open interval (a,b), and f(a) = f(b),
-   then x0 \\<in> (a,b) such that f'(x0) = 0
- ----------------------------------------------------------------------*)
-
-Goal "[| a < b; f(a) = f(b); \
-\        \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\        \\<forall>x. a < x & x < b --> f differentiable x \
-\     |] ==> \\<exists>z. a < z & z < b & DERIV f z :> 0";
-by (ftac (order_less_imp_le RS isCont_eq_Ub) 1);
-by (EVERY1[assume_tac,Step_tac]);
-by (ftac (order_less_imp_le RS isCont_eq_Lb) 1);
-by (EVERY1[assume_tac,Step_tac]);
-by (case_tac "a < x & x < b" 1 THEN etac conjE 1);
-by (Asm_full_simp_tac 2);
-by (forw_inst_tac [("a","a"),("x","x")] lemma_interval 1);
-by (EVERY1[assume_tac,etac exE]);
-by (res_inst_tac [("x","x")] exI 1 THEN Asm_full_simp_tac 1);
-by (subgoal_tac "(\\<exists>l. DERIV f x :> l) & \
-\        (\\<exists>d. 0 < d & (\\<forall>y. abs(x - y) < d --> f(y) \\<le> f(x)))" 1);
-by (Clarify_tac 1 THEN rtac conjI 2);
-by (blast_tac (claset() addIs [differentiableD]) 2);
-by (Blast_tac 2);
-by (ftac DERIV_local_max 1);
-by (EVERY1[Blast_tac,Blast_tac]);
-by (case_tac "a < xa & xa < b" 1 THEN etac conjE 1);
-by (Asm_full_simp_tac 2);
-by (forw_inst_tac [("a","a"),("x","xa")] lemma_interval 1);
-by (EVERY1[assume_tac,etac exE]);
-by (res_inst_tac [("x","xa")] exI 1 THEN Asm_full_simp_tac 1);
-by (subgoal_tac "(\\<exists>l. DERIV f xa :> l) & \
-\        (\\<exists>d. 0 < d & (\\<forall>y. abs(xa - y) < d --> f(xa) \\<le> f(y)))" 1);
-by (Clarify_tac 1 THEN rtac conjI 2);
-by (blast_tac (claset() addIs [differentiableD]) 2);
-by (Blast_tac 2);
-by (ftac DERIV_local_min 1);
-by (EVERY1[Blast_tac,Blast_tac]);
-by (subgoal_tac "\\<forall>x. a \\<le> x & x \\<le> b --> f(x) = f(b)" 1);
-by (Clarify_tac 2);
-by (rtac real_le_anti_sym 2);
-by (subgoal_tac "f b = f x" 2);
-by (Asm_full_simp_tac 2);
-by (res_inst_tac [("x1","a"),("y1","x")] (order_le_imp_less_or_eq RS disjE) 2);
-by (assume_tac 2);
-by (dres_inst_tac [("z","x"),("w","b")] real_le_anti_sym 2);
-by (subgoal_tac "f b = f xa" 5);
-by (Asm_full_simp_tac 5);
-by (res_inst_tac [("x1","a"),("y1","xa")] (order_le_imp_less_or_eq RS disjE) 5);
-by (assume_tac 5);
-by (dres_inst_tac [("z","xa"),("w","b")] real_le_anti_sym 5);
-by (REPEAT(Asm_full_simp_tac 2));
-by (dtac real_dense 1 THEN etac exE 1);
-by (res_inst_tac [("x","r")] exI 1 THEN Asm_simp_tac 1);
-by (etac conjE 1);
-by (forw_inst_tac [("a","a"),("x","r")] lemma_interval 1);
-by (EVERY1[assume_tac, etac exE]);
-by (subgoal_tac "(\\<exists>l. DERIV f r :> l) & \
-\        (\\<exists>d. 0 < d & (\\<forall>y. abs(r - y) < d --> f(r) = f(y)))" 1);
-by (Clarify_tac 1 THEN rtac conjI 2);
-by (blast_tac (claset() addIs [differentiableD]) 2);
-by (EVERY1[ftac DERIV_local_const, Blast_tac, Blast_tac]);
-by (res_inst_tac [("x","d")] exI 1);
-by (EVERY1[rtac conjI, Blast_tac, rtac allI, rtac impI]);
-by (res_inst_tac [("s","f b")] trans 1);
-by (blast_tac (claset() addSDs [order_less_imp_le]) 1);
-by (rtac sym 1 THEN Blast_tac 1);
-qed "Rolle";
-
-(*----------------------------------------------------------------------------*)
-(* Mean value theorem                                                         *)
-(*----------------------------------------------------------------------------*)
-
-Goal "f a - (f b - f a)/(b - a) * a = \
-\     f b - (f b - f a)/(b - a) * (b::real)";
-by (case_tac "a = b" 1);
-by (Asm_full_simp_tac 1);
-by (res_inst_tac [("c1","b - a")] (real_mult_left_cancel RS iffD1) 1);
-by (arith_tac 1);
-by (auto_tac (claset(), simpset() addsimps [right_diff_distrib]));
-by (auto_tac (claset(), simpset() addsimps [left_diff_distrib]));
-qed "lemma_MVT";
-
-Goal "[| a < b; \
-\        \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\        \\<forall>x. a < x & x < b --> f differentiable x |] \
-\     ==>  \\<exists>l z. a < z & z < b & DERIV f z :> l & \
-\                  (f(b) - f(a) = (b - a) * l)";
-by (dres_inst_tac [("f","%x. f(x) - (((f(b) - f(a)) / (b - a)) * x)")]
-    Rolle 1);
-by (rtac lemma_MVT 1);
-by Safe_tac;
-by (rtac isCont_diff 1 THEN Blast_tac 1);
-by (rtac (isCont_const RS isCont_mult) 1);
-by (rtac isCont_Id 1);
-by (dres_inst_tac [("P", "%x. ?Pre x --> f differentiable x"),
-                   ("x","x")] spec 1);
-by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 1);
-by Safe_tac;
-by (res_inst_tac [("x","xa - ((f(b) - f(a)) / (b - a))")] exI 1);
-by (rtac DERIV_diff 1 THEN assume_tac 1);
-(*derivative of a linear function is the constant...*)
-by (subgoal_tac "(%x. (f b - f a) * x / (b - a)) = \
-\                op * ((f b - f a) / (b - a))" 1);
-by (rtac ext 2 THEN Simp_tac 2);
-by (Asm_full_simp_tac 1);
-(*final case*)
-by (res_inst_tac [("x","((f(b) - f(a)) / (b - a))")] exI 1);
-by (res_inst_tac [("x","z")] exI 1);
-by Safe_tac;
-by (Asm_full_simp_tac 2);
-by (subgoal_tac "DERIV (%x. ((f(b) - f(a)) / (b - a)) * x) z :> \
-\                           ((f(b) - f(a)) / (b - a))" 1);
-by (rtac DERIV_cmult_Id 2);
-by (dtac DERIV_add 1 THEN assume_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_assoc, real_diff_def]) 1);
-qed "MVT";
-
-(*----------------------------------------------------------------------------*)
-(* Theorem that function is constant if its derivative is 0 over an interval. *)
-(*----------------------------------------------------------------------------*)
-
-Goal "[| a < b; \
-\        \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\        \\<forall>x. a < x & x < b --> DERIV f x :> 0 |] \
-\       ==> (f b = f a)";
-by (dtac MVT 1 THEN assume_tac 1);
-by (blast_tac (claset() addIs [differentiableI]) 1);
-by (auto_tac (claset() addSDs [DERIV_unique],simpset()
-    addsimps [diff_eq_eq]));
-qed "DERIV_isconst_end";
-
-Goal "[| a < b; \
-\        \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\        \\<forall>x. a < x & x < b --> DERIV f x :> 0 |] \
-\       ==> \\<forall>x. a \\<le> x & x \\<le> b --> f x = f a";
-by Safe_tac;
-by (dres_inst_tac [("x","a")] order_le_imp_less_or_eq 1);
-by Safe_tac;
-by (dres_inst_tac [("b","x")] DERIV_isconst_end 1);
-by Auto_tac;
-qed "DERIV_isconst1";
-
-Goal "[| a < b; \
-\        \\<forall>x. a \\<le> x & x \\<le> b --> isCont f x; \
-\        \\<forall>x. a < x & x < b --> DERIV f x :> 0; \
-\        a \\<le> x; x \\<le> b |] \
-\       ==> f x = f a";
-by (blast_tac (claset() addDs [DERIV_isconst1]) 1);
-qed "DERIV_isconst2";
-
-Goal "\\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)";
-by (res_inst_tac [("x","x"),("y","y")] linorder_cases 1);
-by (rtac sym 1);
-by (auto_tac (claset() addIs [DERIV_isCont,DERIV_isconst_end],simpset()));
-qed "DERIV_isconst_all";
-
-Goal "[|a \\<noteq> b; \\<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b - a) * k";
-by (res_inst_tac [("x","a"),("y","b")] linorder_cases 1);
-by Auto_tac;
-by (ALLGOALS(dres_inst_tac [("f","f")] MVT));
-by (auto_tac (claset() addDs [DERIV_isCont,DERIV_unique],simpset() addsimps
-    [differentiable_def]));
-by (auto_tac (claset() addDs [DERIV_unique],
-       simpset() addsimps [left_distrib, real_diff_def]));
-qed "DERIV_const_ratio_const";
-
-Goal "[|a \\<noteq> b; \\<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b - a) = k";
-by (res_inst_tac [("c1","b - a")] (real_mult_right_cancel RS iffD1) 1);
-by (auto_tac (claset() addSDs [DERIV_const_ratio_const],
-              simpset() addsimps [real_mult_assoc]));
-qed "DERIV_const_ratio_const2";
-
-Goal "((a + b) /2 - a) = (b - a)/(2::real)";
-by Auto_tac;
-qed "real_average_minus_first";
-Addsimps [real_average_minus_first];
-
-Goal "((b + a)/2 - a) = (b - a)/(2::real)";
-by Auto_tac;
-qed "real_average_minus_second";
-Addsimps [real_average_minus_second];
-
-
-(* Gallileo's "trick": average velocity = av. of end velocities *)
-Goal "[|a \\<noteq> (b::real); \\<forall>x. DERIV v x :> k|] \
-\     ==> v((a + b)/2) = (v a + v b)/2";
-by (res_inst_tac [("x","a"),("y","b")] linorder_cases 1);
-by Safe_tac;
-by (ftac DERIV_const_ratio_const2 1 THEN assume_tac 1);
-by (ftac DERIV_const_ratio_const2 2 THEN assume_tac 2);
-by (dtac real_less_half_sum 1);
-by (dtac real_gt_half_sum 2);
-by (ftac (real_not_refl2 RS DERIV_const_ratio_const2) 1 THEN assume_tac 1);
-by (dtac ((real_not_refl2 RS not_sym) RS DERIV_const_ratio_const2) 2
-    THEN assume_tac 2);
-by (ALLGOALS (dres_inst_tac [("f","%u. (b-a)*u")] arg_cong));
-by (auto_tac (claset(), simpset() addsimps [inverse_eq_divide]));
-by (asm_full_simp_tac (simpset() addsimps [real_add_commute, eq_commute]) 1);
-qed "DERIV_const_average";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Dull lemma that an continuous injection on an interval must have a strict*)
-(* maximum at an end point, not in the middle.                              *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[|0 < d; \\<forall>z. abs(z - x) \\<le> d --> g(f z) = z; \
-\       \\<forall>z. abs(z - x) \\<le> d --> isCont f z |]  \
-\     ==> ~(\\<forall>z. abs(z - x) \\<le> d --> f(z) \\<le> f(x))";
-by (rtac notI 1);
-by (rotate_tac 3 1);
-by (forw_inst_tac [("x","x - d")] spec 1);
-by (forw_inst_tac [("x","x + d")] spec 1);
-by Safe_tac;
-by (cut_inst_tac [("x","f(x - d)"),("y","f(x + d)")]
-    (ARITH_PROVE "x \\<le> y | y \\<le> (x::real)") 4);
-by (etac disjE 4);
-by (REPEAT(arith_tac 1));
-by (cut_inst_tac [("f","f"),("a","x - d"),("b","x"),("y","f(x + d)")]
-    IVT_objl 1);
-by Safe_tac;
-by (arith_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (dres_inst_tac [("f","g")] arg_cong 1);
-by (rotate_tac 2 1);
-by (forw_inst_tac [("x","xa")] spec 1);
-by (dres_inst_tac [("x","x + d")] spec 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-(* 2nd case: similar *)
-by (cut_inst_tac [("f","f"),("a","x"),("b","x + d"),("y","f(x - d)")]
-    IVT2_objl 1);
-by Safe_tac;
-by (arith_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (dres_inst_tac [("f","g")] arg_cong 1);
-by (rotate_tac 2 1);
-by (forw_inst_tac [("x","xa")] spec 1);
-by (dres_inst_tac [("x","x - d")] spec 1);
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-qed "lemma_isCont_inj";
-
-(* ------------------------------------------------------------------------ *)
-(* Similar version for lower bound                                          *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[|0 < d; \\<forall>z. abs(z - x) \\<le> d --> g(f z) = z; \
-\       \\<forall>z. abs(z - x) \\<le> d --> isCont f z |]  \
-\     ==> ~(\\<forall>z. abs(z - x) \\<le> d --> f(x) \\<le> f(z))";
-by (auto_tac (claset() addSDs [(asm_full_simplify (simpset())
-    (read_instantiate [("f","%x. - f x"),("g","%y. g(-y)"),("x","x"),("d","d")]
-     lemma_isCont_inj))],simpset() addsimps [isCont_minus]));
-qed "lemma_isCont_inj2";
-
-(* ------------------------------------------------------------------------ *)
-(* Show there's an interval surrounding f(x) in f[[x - d, x + d]]           *)
-(* Also from John's theory                                                  *)
-(* ------------------------------------------------------------------------ *)
-
-val lemma_le = ARITH_PROVE "0 \\<le> (d::real) ==> -d \\<le> d";
-
-(* FIXME: awful proof - needs improvement *)
-Goal "[| 0 < d; \\<forall>z. abs(z - x) \\<le> d --> g(f z) = z; \
-\        \\<forall>z. abs(z - x) \\<le> d --> isCont f z |] \
-\      ==> \\<exists>e. 0 < e & \
-\                 (\\<forall>y. \
-\                     abs(y - f(x)) \\<le> e --> \
-\                     (\\<exists>z. abs(z - x) \\<le> d & (f z = y)))";
-by (ftac order_less_imp_le 1);
-by (dtac (lemma_le RS (asm_full_simplify (simpset()) (read_instantiate
-    [("f","f"),("a","x - d"),("b","x + d")] isCont_Lb_Ub))) 1);
-by Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (subgoal_tac "L \\<le> f x & f x \\<le> M" 1);
-by (dres_inst_tac [("P", "%v. ?P v --> ?Q v & ?R v"), ("x","x")] spec 2);
-by (Asm_full_simp_tac 2);
-by (subgoal_tac "L < f x & f x < M" 1);
-by Safe_tac;
-by (dres_inst_tac [("x","L")] (ARITH_PROVE "x < y ==> 0 < y - (x::real)") 1);
-by (dres_inst_tac [("x","f x")] (ARITH_PROVE "x < y ==> 0 < y - (x::real)") 1);
-by (dres_inst_tac [("d1.0","f x - L"),("d2.0","M - f x")]
-    (real_lbound_gt_zero) 1);
-by Safe_tac;
-by (res_inst_tac [("x","e")] exI 1);
-by Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps [abs_le_interval_iff]) 1);
-by (dres_inst_tac [("P","%v. ?PP v --> (\\<exists>xa. ?Q v xa)"),("x","y")] spec 1);
-by (Step_tac 1 THEN REPEAT(arith_tac 1));
-by (res_inst_tac [("x","xa")] exI 1);
-by (arith_tac 1);
-by (ALLGOALS(etac (ARITH_PROVE "[|x \\<le> y; x \\<noteq> y |] ==> x < (y::real)")));
-by (ALLGOALS(rotate_tac 3));
-by (dtac lemma_isCont_inj2 1);
-by (assume_tac 2);
-by (dtac lemma_isCont_inj 3);
-by (assume_tac 4);
-by (TRYALL(assume_tac));
-by Safe_tac;
-by (ALLGOALS(dres_inst_tac [("x","z")] spec));
-by (ALLGOALS(arith_tac));
-qed "isCont_inj_range";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Continuity of inverse function                                           *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[| 0 < d; \\<forall>z. abs(z - x) \\<le> d --> g(f(z)) = z; \
-\        \\<forall>z. abs(z - x) \\<le> d --> isCont f z |] \
-\     ==> isCont g (f x)";
-by (simp_tac (simpset() addsimps [isCont_iff,LIM_def]) 1);
-by Safe_tac;
-by (dres_inst_tac [("d1.0","r")] (real_lbound_gt_zero) 1);
-by (assume_tac 1 THEN Step_tac 1);
-by (subgoal_tac "\\<forall>z. abs(z - x) \\<le> e --> (g(f z) = z)" 1);
-by (Force_tac 2);
-by (subgoal_tac "\\<forall>z. abs(z - x) \\<le> 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 Safe_tac;
-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_inverse_function";
-
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/Lim.thy
--- a/src/HOL/Hyperreal/Lim.thy	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/Lim.thy	Fri Mar 19 10:51:03 2004 +0100
@@ -1,66 +1,67 @@
 (*  Title       : Lim.thy
+    ID          : $Id$
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
-    Description : Theory of limits, continuity and 
-                  differentiation of real=>real functions
+    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
 *)
 
-Lim = SEQ + RealDef + 
+header{*Limits, Continuity and Differentiation*}
 
-(*-----------------------------------------------------------------------
-    Limits, continuity and differentiation: standard and NS definitions
- -----------------------------------------------------------------------*)
+theory Lim = SEQ + RealDef:
+
+text{*Standard and Nonstandard Definitions*}
 
 constdefs
-  LIM :: [real=>real,real,real] => bool
+  LIM :: "[real=>real,real,real] => bool"
 				("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60)
   "f -- a --> L ==
-     ALL r. 0 < r --> 
-	     (EX s. 0 < s & (ALL x. (x ~= a & (abs(x + -a) < s)
-			  --> abs(f x + -L) < r)))"
+     \<forall>r. 0 < r -->
+	     (\<exists>s. 0 < s & (\<forall>x. (x \<noteq> a & (\<bar>x + -a\<bar> < s)
+			  --> \<bar>f x + -L\<bar> < r)))"
 
-  NSLIM :: [real=>real,real,real] => bool
+  NSLIM :: "[real=>real,real,real] => bool"
 			      ("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60)
-  "f -- a --NS> L == (ALL x. (x ~= hypreal_of_real a & 
+  "f -- a --NS> L == (\<forall>x. (x \<noteq> hypreal_of_real a &
 		      x @= hypreal_of_real a -->
-		      ( *f* f) x @= hypreal_of_real L))"   
+		      ( *f* f) x @= hypreal_of_real L))"
 
-  isCont :: [real=>real,real] => bool
-  "isCont f a == (f -- a --> (f a))"        
+  isCont :: "[real=>real,real] => bool"
+  "isCont f a == (f -- a --> (f a))"
 
   (* NS definition dispenses with limit notions *)
-  isNSCont :: [real=>real,real] => bool
-  "isNSCont f a == (ALL y. y @= hypreal_of_real a --> 
+  isNSCont :: "[real=>real,real] => bool"
+  "isNSCont f a == (\<forall>y. y @= hypreal_of_real a -->
 			   ( *f* f) y @= hypreal_of_real (f a))"
 
   (* differentiation: D is derivative of function f at x *)
-  deriv:: [real=>real,real,real] => bool
+  deriv:: "[real=>real,real,real] => bool"
 			    ("(DERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
-  "DERIV f x :> D == ((%h. (f(x + h) + -f(x))/h) -- 0 --> D)"
+  "DERIV f x :> D == ((%h. (f(x + h) + -f x)/h) -- 0 --> D)"
 
-  nsderiv :: [real=>real,real,real] => bool
+  nsderiv :: "[real=>real,real,real] => bool"
 			    ("(NSDERIV (_)/ (_)/ :> (_))" [60, 0, 60] 60)
-  "NSDERIV f x :> D == (ALL h: Infinitesimal - {0}. 
-			(( *f* f)(hypreal_of_real x + h) + 
+  "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)"
 
-  differentiable :: [real=>real,real] => bool   (infixl 60)
-  "f differentiable x == (EX D. DERIV f x :> D)"
+  differentiable :: "[real=>real,real] => bool"   (infixl "differentiable" 60)
+  "f differentiable x == (\<exists>D. DERIV f x :> D)"
 
-  NSdifferentiable :: [real=>real,real] => bool   (infixl 60)
-  "f NSdifferentiable x == (EX D. NSDERIV f x :> D)"
+  NSdifferentiable :: "[real=>real,real] => bool"   
+                       (infixl "NSdifferentiable" 60)
+  "f NSdifferentiable x == (\<exists>D. NSDERIV f x :> D)"
 
-  increment :: [real=>real,real,hypreal] => hypreal
-  "increment f x h == (@inc. f NSdifferentiable x & 
+  increment :: "[real=>real,real,hypreal] => hypreal"
+  "increment f x h == (@inc. f NSdifferentiable x &
 		       inc = ( *f* f)(hypreal_of_real x + h) + -hypreal_of_real (f x))"
 
-  isUCont :: (real=>real) => bool
-  "isUCont f ==  (ALL r. 0 < r --> 
-		      (EX s. 0 < s & (ALL x y. abs(x + -y) < s
-			    --> abs(f x + -f y) < r)))"
+  isUCont :: "(real=>real) => bool"
+  "isUCont f ==  (\<forall>r. 0 < r -->
+		      (\<exists>s. 0 < s & (\<forall>x y. \<bar>x + -y\<bar> < s
+			    --> \<bar>f x + -f y\<bar> < r)))"
 
-  isNSUCont :: (real=>real) => bool
-  "isNSUCont f == (ALL x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
+  isNSUCont :: "(real=>real) => bool"
+  "isNSUCont f == (\<forall>x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
 
 
 (*Used in the proof of the Bolzano theorem*)
@@ -72,8 +73,2258 @@
   "Bolzano_bisect P a b (Suc n) =
       (let (x,y) = Bolzano_bisect P a b n
        in if P(x, (x+y)/2) then ((x+y)/2, y)
-                            else (x, (x+y)/2) )"
-  
+                            else (x, (x+y)/2))"
+
+
+
+section{*Some Purely Standard Proofs*}
+
+lemma LIM_eq:
+     "f -- a --> L =
+     (\<forall>r. 0<r --> (\<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)))"
+by (simp add: LIM_def diff_minus)
+
+lemma LIM_D:
+     "[| f -- a --> L; 0<r |]
+      ==> \<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)"
+by (simp add: LIM_eq)
+
+lemma LIM_const: "(%x. k) -- x --> k"
+by (simp add: LIM_def)
+declare LIM_const [simp]
+
+lemma LIM_add:
+  assumes f: "f -- a --> L" and g: "g -- a --> M"
+  shows "(%x. f x + g(x)) -- a --> (L + M)"
+proof (simp add: LIM_eq, clarify)
+  fix r :: real
+  assume r: "0<r"
+  from LIM_D [OF f half_gt_zero [OF r]]
+  obtain fs
+    where fs:    "0 < fs"
+      and fs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < fs --> \<bar>f x - L\<bar> < r/2"
+  by blast
+  from LIM_D [OF g half_gt_zero [OF r]]
+  obtain gs
+    where gs:    "0 < gs"
+      and gs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < gs --> \<bar>g x - M\<bar> < r/2"
+  by blast
+  show "\<exists>s. 0 < s \<and>
+            (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> \<bar>f x + g x - (L + M)\<bar> < r)"
+  proof (intro exI conjI strip)
+    show "0 < min fs gs"  by (simp add: fs gs)
+    fix x :: real
+    assume "x \<noteq> a \<and> \<bar>x-a\<bar> < min fs gs"
+    with fs_lt gs_lt
+    have "\<bar>f x - L\<bar> < r/2" and "\<bar>g x - M\<bar> < r/2" by (auto simp add: fs_lt)
+    hence "\<bar>f x - L\<bar> + \<bar>g x - M\<bar> < r" by arith
+    thus "\<bar>f x + g x - (L + M)\<bar> < r"
+      by (blast intro: abs_diff_triangle_ineq order_le_less_trans)
+  qed
+qed
+
+lemma LIM_minus: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
+apply (simp add: LIM_eq)
+apply (subgoal_tac "\<forall>x. \<bar>- f x + L\<bar> = \<bar>f x - L\<bar>")
+apply (simp_all add: abs_if)
+done
+
+lemma LIM_add_minus:
+    "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
+by (blast dest: LIM_add LIM_minus)
+
+lemma LIM_diff:
+    "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) - g(x)) -- x --> l-m"
+by (simp add: diff_minus LIM_add_minus) 
+
+
+lemma LIM_const_not_eq: "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
+proof (simp add: linorder_neq_iff LIM_eq, elim disjE)
+  assume k: "k < L"
+  show "\<exists>r. 0 < r \<and>
+        (\<forall>s. 0 < s \<longrightarrow> (\<exists>x. (x < a \<or> a < x) \<and> \<bar>x-a\<bar> < s) \<and> \<not> \<bar>k-L\<bar> < r)"
+  proof (intro exI conjI strip)
+    show "0 < L-k" by (simp add: k)
+    fix s :: real
+    assume s: "0<s"
+    { from s show "s/2 + a < a \<or> a < s/2 + a" by arith
+     next
+      from s show "\<bar>s / 2 + a - a\<bar> < s" by (simp add: abs_if) 
+     next
+      from s show "~ \<bar>k-L\<bar> < L-k" by (simp add: abs_if) }
+  qed
+next
+  assume k: "L < k"
+  show "\<exists>r. 0 < r \<and>
+        (\<forall>s. 0 < s \<longrightarrow> (\<exists>x. (x < a \<or> a < x) \<and> \<bar>x-a\<bar> < s) \<and> \<not> \<bar>k-L\<bar> < r)"
+  proof (intro exI conjI strip)
+    show "0 < k-L" by (simp add: k)
+    fix s :: real
+    assume s: "0<s"
+    { from s show "s/2 + a < a \<or> a < s/2 + a" by arith
+     next
+      from s show "\<bar>s / 2 + a - a\<bar> < s" by (simp add: abs_if) 
+     next
+      from s show "~ \<bar>k-L\<bar> < k-L" by (simp add: abs_if) }
+  qed
+qed
+
+lemma LIM_const_eq: "(%x. k) -- x --> L ==> k = L"
+apply (rule ccontr)
+apply (blast dest: LIM_const_not_eq) 
+done
+
+lemma LIM_unique: "[| f -- a --> L; f -- a --> M |] ==> L = M"
+apply (drule LIM_diff, assumption) 
+apply (auto dest!: LIM_const_eq)
+done
+
+lemma LIM_mult_zero:
+  assumes f: "f -- a --> 0" and g: "g -- a --> 0"
+  shows "(%x. f(x) * g(x)) -- a --> 0"
+proof (simp add: LIM_eq, clarify)
+  fix r :: real
+  assume r: "0<r"
+  from LIM_D [OF f zero_less_one]
+  obtain fs
+    where fs:    "0 < fs"
+      and fs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < fs --> \<bar>f x\<bar> < 1"
+  by auto
+  from LIM_D [OF g r]
+  obtain gs
+    where gs:    "0 < gs"
+      and gs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < gs --> \<bar>g x\<bar> < r"
+  by auto
+  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> \<bar>f x\<bar> * \<bar>g x\<bar> < r)"
+  proof (intro exI conjI strip)
+    show "0 < min fs gs"  by (simp add: fs gs)
+    fix x :: real
+    assume "x \<noteq> a \<and> \<bar>x-a\<bar> < min fs gs"
+    with fs_lt gs_lt
+    have "\<bar>f x\<bar> < 1" and "\<bar>g x\<bar> < r" by (auto simp add: fs_lt)
+    hence "\<bar>f x\<bar> * \<bar>g x\<bar> < 1*r" by (rule abs_mult_less) 
+    thus "\<bar>f x\<bar> * \<bar>g x\<bar> < r" by simp
+  qed
+qed
+
+lemma LIM_self: "(%x. x) -- a --> a"
+by (auto simp add: LIM_def)
+
+text{*Limits are equal for functions equal except at limit point*}
+lemma LIM_equal:
+     "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
+by (simp add: LIM_def)
+
+text{*Two uses in Hyperreal/Transcendental.ML*}
+lemma LIM_trans:
+     "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
+apply (drule LIM_add, assumption)
+apply (auto simp add: add_assoc)
+done
+
+
+subsection{*Relationships Between Standard and Nonstandard Concepts*}
+
+text{*Standard and NS definitions of Limit*} (*NEEDS STRUCTURING*)
+lemma LIM_NSLIM:
+      "f -- x --> L ==> f -- x --NS> L"
+apply (simp add: LIM_def NSLIM_def approx_def)
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
+apply (rule_tac z = xa in eq_Abs_hypreal)
+apply (auto simp add: real_add_minus_iff starfun hypreal_minus hypreal_of_real_def hypreal_add)
+apply (rule bexI, rule_tac [2] lemma_hyprel_refl, clarify) 
+apply (drule_tac x = u in spec, clarify)
+apply (drule_tac x = s in spec, clarify)
+apply (subgoal_tac "\<forall>n::nat. (xa n) \<noteq> x & abs ((xa n) + - x) < s --> abs (f (xa n) + - L) < u")
+prefer 2 apply blast
+apply (drule FreeUltrafilterNat_all, ultra)
+done
+
+(*---------------------------------------------------------------------
+    Limit: NS definition ==> standard definition
+ ---------------------------------------------------------------------*)
+
+lemma lemma_LIM: "\<forall>s. 0 < s --> (\<exists>xa.  xa \<noteq> x &
+         \<bar>xa + - x\<bar> < s  & r \<le> \<bar>f xa + -L\<bar>)
+      ==> \<forall>n::nat. \<exists>xa.  xa \<noteq> x &
+              \<bar>xa + -x\<bar> < inverse(real(Suc n)) & r \<le> \<bar>f xa + -L\<bar>"
+apply clarify
+apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
+done
+
+lemma lemma_skolemize_LIM2:
+     "\<forall>s. 0 < s --> (\<exists>xa.  xa \<noteq> x &
+         \<bar>xa + - x\<bar> < s  & r \<le> \<bar>f xa + -L\<bar>)
+      ==> \<exists>X. \<forall>n::nat. X n \<noteq> x &
+                \<bar>X n + -x\<bar> < inverse(real(Suc n)) & r \<le> abs(f (X n) + -L)"
+apply (drule lemma_LIM)
+apply (drule choice, blast)
+done
+
+lemma lemma_simp: "\<forall>n. X n \<noteq> x &
+          \<bar>X n + - x\<bar> < inverse (real(Suc n)) &
+          r \<le> abs (f (X n) + - L) ==>
+          \<forall>n. \<bar>X n + - x\<bar> < inverse (real(Suc n))"
+by auto
+
+
+(*-------------------
+    NSLIM => LIM
+ -------------------*)
+
+lemma NSLIM_LIM: "f -- x --NS> L ==> f -- x --> L"
+apply (simp add: LIM_def NSLIM_def approx_def)
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, clarify)
+apply (rule ccontr, simp)  
+apply (simp add: linorder_not_less)
+apply (drule lemma_skolemize_LIM2, safe)
+apply (drule_tac x = "Abs_hypreal (hyprel``{X}) " in spec)
+apply (auto simp add: starfun hypreal_minus hypreal_of_real_def hypreal_add)
+apply (drule lemma_simp [THEN real_seq_to_hypreal_Infinitesimal])
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff hypreal_of_real_def hypreal_minus hypreal_add, blast)
+apply (drule spec, drule mp, assumption)
+apply (drule FreeUltrafilterNat_all, ultra)
+done
+
+
+(**** Key result ****)
+lemma LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
+by (blast intro: LIM_NSLIM NSLIM_LIM)
+
+(*-------------------------------------------------------------------*)
+(*   Proving properties of limits using nonstandard definition and   *)
+(*   hence, the properties hold for standard limits as well          *)
+(*-------------------------------------------------------------------*)
+(*------------------------------------------------
+      NSLIM_mult and hence (trivially) LIM_mult
+ ------------------------------------------------*)
+
+lemma NSLIM_mult:
+     "[| f -- x --NS> l; g -- x --NS> m |]
+      ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
+apply (simp add: NSLIM_def)
+apply (auto intro!: approx_mult_HFinite)
+done
+
+lemma LIM_mult2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) * g(x)) -- x --> (l * m)"
+by (simp add: LIM_NSLIM_iff NSLIM_mult)
+
+(*----------------------------------------------
+      NSLIM_add and hence (trivially) LIM_add
+      Note the much shorter proof
+ ----------------------------------------------*)
+lemma NSLIM_add:
+     "[| f -- x --NS> l; g -- x --NS> m |]
+      ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
+apply (simp add: NSLIM_def)
+apply (auto intro!: approx_add)
+done
+
+lemma LIM_add2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + g(x)) -- x --> (l + m)"
+by (simp add: LIM_NSLIM_iff NSLIM_add)
+
+
+lemma NSLIM_const: "(%x. k) -- x --NS> k"
+by (simp add: NSLIM_def)
+
+declare NSLIM_const [simp]
+
+lemma LIM_const2: "(%x. k) -- x --> k"
+by (simp add: LIM_NSLIM_iff)
+
+
+lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
+by (simp add: NSLIM_def)
+
+lemma LIM_minus2: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
+by (simp add: LIM_NSLIM_iff NSLIM_minus)
+
+
+lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
+by (blast dest: NSLIM_add NSLIM_minus)
+
+lemma LIM_add_minus2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
+by (simp add: LIM_NSLIM_iff NSLIM_add_minus)
+
+
+lemma NSLIM_inverse:
+     "[| f -- a --NS> L;  L \<noteq> 0 |]
+      ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
+apply (simp add: NSLIM_def, clarify)
+apply (drule spec)
+apply (auto simp add: hypreal_of_real_approx_inverse)
+done
+
+lemma LIM_inverse: "[| f -- a --> L; L \<noteq> 0 |] ==> (%x. inverse(f(x))) -- a --> (inverse L)"
+by (simp add: LIM_NSLIM_iff NSLIM_inverse)
+
+
+lemma NSLIM_zero:
+  assumes f: "f -- a --NS> l" shows "(%x. f(x) + -l) -- a --NS> 0"
+proof -;
+  have "(\<lambda>x. f x + - l) -- a --NS> l + -l"
+    by (rule NSLIM_add_minus [OF f NSLIM_const]) 
+  thus ?thesis by simp
+qed
+
+lemma LIM_zero2: "f -- a --> l ==> (%x. f(x) + -l) -- a --> 0"
+by (simp add: LIM_NSLIM_iff NSLIM_zero)
+
+lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
+apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
+apply (auto simp add: diff_minus add_assoc)
+done
+
+lemma LIM_zero_cancel: "(%x. f(x) - l) -- x --> 0 ==> f -- x --> l"
+apply (drule_tac g = "%x. l" and M = l in LIM_add)
+apply (auto simp add: diff_minus add_assoc)
+done
+
+
+
+lemma NSLIM_not_zero: "k \<noteq> 0 ==> ~ ((%x. k) -- x --NS> 0)"
+apply (simp add: NSLIM_def)
+apply (rule_tac x = "hypreal_of_real x + epsilon" in exI)
+apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
+            simp add: hypreal_epsilon_not_zero)
+done
+
+lemma NSLIM_const_not_eq: "k \<noteq> L ==> ~ ((%x. k) -- x --NS> L)"
+apply (simp add: NSLIM_def)
+apply (rule_tac x = "hypreal_of_real x + epsilon" in exI)
+apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
+            simp add: hypreal_epsilon_not_zero)
+done
+
+lemma NSLIM_const_eq: "(%x. k) -- x --NS> L ==> k = L"
+apply (rule ccontr)
+apply (blast dest: NSLIM_const_not_eq) 
+done
+
+(* can actually be proved more easily by unfolding def! *)
+lemma NSLIM_unique: "[| f -- x --NS> L; f -- x --NS> M |] ==> L = M"
+apply (drule NSLIM_minus)
+apply (drule NSLIM_add, assumption)
+apply (auto dest!: NSLIM_const_eq [symmetric])
+done
+
+lemma LIM_unique2: "[| f -- x --> L; f -- x --> M |] ==> L = M"
+by (simp add: LIM_NSLIM_iff NSLIM_unique)
+
+
+lemma NSLIM_mult_zero: "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
+by (drule NSLIM_mult, auto)
+
+(* we can use the corresponding thm LIM_mult2 *)
+(* for standard definition of limit           *)
+
+lemma LIM_mult_zero2: "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0"
+by (drule LIM_mult2, auto)
+
+
+lemma NSLIM_self: "(%x. x) -- a --NS> a"
+by (simp add: NSLIM_def)
+
+
+(*-----------------------------------------------------------------------------
+   Derivatives and Continuity - NS and Standard properties
+ -----------------------------------------------------------------------------*)
+text{*Continuity*}
+
+lemma isNSContD: "[| isNSCont f a; y \<approx> hypreal_of_real a |] ==> ( *f* f) y \<approx> hypreal_of_real (f a)"
+by (simp add: isNSCont_def)
+
+lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
+by (simp add: isNSCont_def NSLIM_def)
+
+lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
+apply (simp add: isNSCont_def NSLIM_def, auto)
+apply (rule_tac Q = "y = hypreal_of_real a" in excluded_middle [THEN disjE], auto)
+done
+
+(*-----------------------------------------------------
+    NS continuity can be defined using NS Limit in
+    similar fashion to standard def of continuity
+ -----------------------------------------------------*)
+lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
+by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
+
+(*----------------------------------------------
+  Hence, NS continuity can be given
+  in terms of standard limit
+ ---------------------------------------------*)
+lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
+by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
+
+(*-----------------------------------------------
+  Moreover, it's trivial now that NS continuity
+  is equivalent to standard continuity
+ -----------------------------------------------*)
+lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
+apply (simp add: isCont_def)
+apply (rule isNSCont_LIM_iff)
+done
+
+(*----------------------------------------
+  Standard continuity ==> NS continuity
+ ----------------------------------------*)
+lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
+by (erule isNSCont_isCont_iff [THEN iffD2])
+
+(*----------------------------------------
+  NS continuity ==> Standard continuity
+ ----------------------------------------*)
+lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
+by (erule isNSCont_isCont_iff [THEN iffD1])
+
+text{*Alternative definition of continuity*}
+(* Prove equivalence between NS limits - *)
+(* seems easier than using standard def  *)
+lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
+apply (simp add: NSLIM_def, auto)
+apply (drule_tac x = "hypreal_of_real a + x" in spec)
+apply (drule_tac [2] x = "-hypreal_of_real a + x" in spec, safe, simp)
+apply (rule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
+apply (rule_tac [4] approx_minus_iff2 [THEN iffD1])
+ prefer 3 apply (simp add: add_commute) 
+apply (rule_tac [2] z = x in eq_Abs_hypreal)
+apply (rule_tac [4] z = x in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_of_real_def hypreal_minus hypreal_add add_assoc approx_refl hypreal_zero_def)
+done
+
+lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
+by (rule NSLIM_h_iff)
+
+lemma LIM_isCont_iff: "(f -- a --> f a) = ((%h. f(a + h)) -- 0 --> f(a))"
+by (simp add: LIM_NSLIM_iff NSLIM_isCont_iff)
+
+lemma isCont_iff: "(isCont f x) = ((%h. f(x + h)) -- 0 --> f(x))"
+by (simp add: isCont_def LIM_isCont_iff)
+
+(*--------------------------------------------------------------------------
+   Immediate application of nonstandard criterion for continuity can offer
+   very simple proofs of some standard property of continuous functions
+ --------------------------------------------------------------------------*)
+text{*sum continuous*}
+lemma isCont_add: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a"
+by (auto intro: approx_add simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
+
+text{*mult continuous*}
+lemma isCont_mult: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a"
+by (auto intro!: starfun_mult_HFinite_approx 
+            simp del: starfun_mult [symmetric] 
+            simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
+
+(*-------------------------------------------
+     composition of continuous functions
+     Note very short straightforard proof!
+ ------------------------------------------*)
+lemma isCont_o: "[| isCont f a; isCont g (f a) |] ==> isCont (g o f) a"
+by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_def starfun_o [symmetric])
+
+lemma isCont_o2: "[| isCont f a; isCont g (f a) |] ==> isCont (%x. g (f x)) a"
+by (auto dest: isCont_o simp add: o_def)
+
+lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
+by (simp add: isNSCont_def)
+
+lemma isCont_minus: "isCont f a ==> isCont (%x. - f x) a"
+by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_minus)
+
+lemma isCont_inverse:
+      "[| isCont f x; f x \<noteq> 0 |] ==> isCont (%x. inverse (f x)) x"
+apply (simp add: isCont_def)
+apply (blast intro: LIM_inverse)
+done
+
+lemma isNSCont_inverse: "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
+by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
+
+lemma isCont_diff:
+      "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a"
+apply (simp add: diff_minus)
+apply (auto intro: isCont_add isCont_minus)
+done
+
+lemma isCont_const: "isCont (%x. k) a"
+by (simp add: isCont_def)
+declare isCont_const [simp]
+
+lemma isNSCont_const: "isNSCont (%x. k) a"
+by (simp add: isNSCont_def)
+declare isNSCont_const [simp]
+
+lemma isNSCont_rabs: "isNSCont abs a"
+apply (simp add: isNSCont_def)
+apply (auto intro: approx_hrabs simp add: hypreal_of_real_hrabs [symmetric] starfun_rabs_hrabs)
+done
+declare isNSCont_rabs [simp]
+
+lemma isCont_rabs: "isCont abs a"
+by (auto simp add: isNSCont_isCont_iff [symmetric])
+declare isCont_rabs [simp]
+
+(****************************************************************
+(%* Leave as commented until I add topology theory or remove? *%)
+(%*------------------------------------------------------------
+  Elementary topology proof for a characterisation of
+  continuity now: a function f is continuous if and only
+  if the inverse image, {x. f(x) \<in> A}, of any open set A
+  is always an open set
+ ------------------------------------------------------------*%)
+Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
+               ==> isNSopen {x. f x \<in> A}"
+by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
+by (dtac (mem_monad_approx RS approx_sym);
+by (dres_inst_tac [("x","a")] spec 1);
+by (dtac isNSContD 1 THEN assume_tac 1)
+by (dtac bspec 1 THEN assume_tac 1)
+by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
+by (blast_tac (claset() addIs [starfun_mem_starset]);
+qed "isNSCont_isNSopen";
+
+Goalw [isNSCont_def]
+          "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
+\              ==> isNSCont f x";
+by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
+     (approx_minus_iff RS iffD2)],simpset() addsimps
+      [Infinitesimal_def,SReal_iff]));
+by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
+by (etac (isNSopen_open_interval RSN (2,impE));
+by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
+by (dres_inst_tac [("x","x")] spec 1);
+by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
+    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
+qed "isNSopen_isNSCont";
+
+Goal "(\<forall>x. isNSCont f x) = \
+\     (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
+by (blast_tac (claset() addIs [isNSCont_isNSopen,
+    isNSopen_isNSCont]);
+qed "isNSCont_isNSopen_iff";
+
+(%*------- Standard version of same theorem --------*%)
+Goal "(\<forall>x. isCont f x) = \
+\         (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
+by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
+              simpset() addsimps [isNSopen_isopen_iff RS sym,
+              isNSCont_isCont_iff RS sym]));
+qed "isCont_isopen_iff";
+*******************************************************************)
+
+text{*Uniform continuity*}
+lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
+by (simp add: isNSUCont_def)
+
+lemma isUCont_isCont: "isUCont f ==> isCont f x"
+by (simp add: isUCont_def isCont_def LIM_def, meson)
+
+lemma isUCont_isNSUCont: "isUCont f ==> isNSUCont f"
+apply (simp add: isNSUCont_def isUCont_def approx_def)
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
+apply (rule_tac z = x in eq_Abs_hypreal)
+apply (rule_tac z = y in eq_Abs_hypreal)
+apply (auto simp add: starfun hypreal_minus hypreal_add)
+apply (rule bexI, rule_tac [2] lemma_hyprel_refl, safe)
+apply (drule_tac x = u in spec, clarify)
+apply (drule_tac x = s in spec, clarify)
+apply (subgoal_tac "\<forall>n::nat. abs ((xa n) + - (xb n)) < s --> abs (f (xa n) + - f (xb n)) < u")
+prefer 2 apply blast
+apply (erule_tac V = "\<forall>x y. \<bar>x + - y\<bar> < s --> \<bar>f x + - f y\<bar> < u" in thin_rl)
+apply (drule FreeUltrafilterNat_all, ultra)
+done
+
+lemma lemma_LIMu: "\<forall>s. 0 < s --> (\<exists>z y. \<bar>z + - y\<bar> < s & r \<le> \<bar>f z + -f y\<bar>)
+      ==> \<forall>n::nat. \<exists>z y.
+               \<bar>z + -y\<bar> < inverse(real(Suc n)) &
+               r \<le> \<bar>f z + -f y\<bar>"
+apply clarify
+apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
+done
+
+lemma lemma_skolemize_LIM2u: "\<forall>s. 0 < s --> (\<exists>z y. \<bar>z + - y\<bar> < s  & r \<le> \<bar>f z + -f y\<bar>)
+      ==> \<exists>X Y. \<forall>n::nat.
+               abs(X n + -(Y n)) < inverse(real(Suc n)) &
+               r \<le> abs(f (X n) + -f (Y n))"
+apply (drule lemma_LIMu)
+apply (drule choice, safe)
+apply (drule choice, blast)
+done
+
+lemma lemma_simpu: "\<forall>n. \<bar>X n + -Y n\<bar> < inverse (real(Suc n)) &
+          r \<le> abs (f (X n) + - f(Y n)) ==>
+          \<forall>n. \<bar>X n + - Y n\<bar> < inverse (real(Suc n))"
+apply auto
+done
+
+lemma isNSUCont_isUCont:
+     "isNSUCont f ==> isUCont f"
+apply (simp add: isNSUCont_def isUCont_def approx_def)
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
+apply (rule ccontr, simp) 
+apply (simp add: linorder_not_less)
+apply (drule lemma_skolemize_LIM2u, safe)
+apply (drule_tac x = "Abs_hypreal (hyprel``{X}) " in spec)
+apply (drule_tac x = "Abs_hypreal (hyprel``{Y}) " in spec)
+apply (simp add: starfun hypreal_minus hypreal_add, auto)
+apply (drule lemma_simpu [THEN real_seq_to_hypreal_Infinitesimal2])
+apply (simp add: Infinitesimal_FreeUltrafilterNat_iff hypreal_minus hypreal_add, blast)
+apply (rotate_tac 2)
+apply (drule_tac x = r in spec, clarify)
+apply (drule FreeUltrafilterNat_all, ultra)
+done
+
+text{*Derivatives*}
+lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --> D)"
+by (simp add: deriv_def)
+
+lemma DERIV_NS_iff:
+      "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --NS> D)"
+by (simp add: deriv_def LIM_NSLIM_iff)
+
+lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) + - f(x))/h) -- 0 --> D"
+by (simp add: deriv_def)
+
+lemma NS_DERIV_D: "DERIV f x :> D ==>
+           (%h. (f(x + h) + - f(x))/h) -- 0 --NS> D"
+by (simp add: deriv_def LIM_NSLIM_iff)
+
+subsubsection{*Uniqueness*}
+
+lemma DERIV_unique:
+      "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
+apply (simp add: deriv_def)
+apply (blast intro: LIM_unique)
+done
+
+lemma NSDeriv_unique:
+     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
+apply (simp add: nsderiv_def)
+apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
+apply (auto dest!: bspec [where x=epsilon] 
+            intro!: inj_hypreal_of_real [THEN injD] 
+            dest: approx_trans3)
+done
+
+subsubsection{*Differentiable*}
+
+lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
+by (simp add: differentiable_def)
+
+lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
+by (force simp add: differentiable_def)
+
+lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
+by (simp add: NSdifferentiable_def)
+
+lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
+by (force simp add: NSdifferentiable_def)
+
+subsubsection{*Alternative definition for differentiability*}
+
+lemma LIM_I:
+     "(!!r. 0<r ==> (\<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)))
+      ==> f -- a --> L"
+by (simp add: LIM_eq)
+
+lemma DERIV_LIM_iff:
+     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
+      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
+proof (intro iffI LIM_I)
+  fix r::real
+  assume r: "0<r"
+  assume "(\<lambda>h. (f (a + h) - f a) / h) -- 0 --> D"
+  from LIM_D [OF this r]
+  obtain s
+    where s:    "0 < s"
+      and s_lt: "\<forall>x. x \<noteq> 0 & \<bar>x\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r"
+  by auto
+  show "\<exists>s. 0 < s \<and>
+            (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> \<bar>(f x - f a) / (x-a) - D\<bar> < r)"
+  proof (intro exI conjI strip)
+    show "0 < s"  by (rule s)
+  next
+    fix x::real
+    assume "x \<noteq> a \<and> \<bar>x-a\<bar> < s"
+    with s_lt [THEN spec [where x="x-a"]]
+    show "\<bar>(f x - f a) / (x-a) - D\<bar> < r" by auto
+  qed
+next
+  fix r::real
+  assume r: "0<r"
+  assume "(\<lambda>x. (f x - f a) / (x-a)) -- a --> D"
+  from LIM_D [OF this r]
+  obtain s
+    where s:    "0 < s"
+      and s_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>(f x - f a)/(x-a) - D\<bar> < r"
+  by auto
+  show "\<exists>s. 0 < s \<and>
+            (\<forall>x. x \<noteq> 0 & \<bar>x - 0\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r)"
+  proof (intro exI conjI strip)
+    show "0 < s"  by (rule s)
+  next
+    fix x::real
+    assume "x \<noteq> 0 \<and> \<bar>x - 0\<bar> < s"
+    with s_lt [THEN spec [where x="x+a"]]
+    show "\<bar>(f (a + x) - f a) / x - D\<bar> < r" by (auto simp add: add_ac)
+  qed
+qed
+
+lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
+by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
+
+
+subsection{*Equivalence of NS and standard definitions of differentiation*}
+
+text{*First NSDERIV in terms of NSLIM*}
+
+(*--- first equivalence ---*)
+lemma NSDERIV_NSLIM_iff:
+      "(NSDERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --NS> D)"
+apply (simp add: nsderiv_def NSLIM_def, auto)
+apply (drule_tac x = xa in bspec)
+apply (rule_tac [3] ccontr)
+apply (drule_tac [3] x = h in spec)
+apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
+done
+
+(*--- second equivalence ---*)
+lemma NSDERIV_NSLIM_iff2:
+     "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
+by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff  diff_minus [symmetric] 
+              LIM_NSLIM_iff [symmetric])
+
+(* while we're at it! *)
+lemma NSDERIV_iff2:
+     "(NSDERIV f x :> D) =
+      (\<forall>w.
+        w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
+        ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
+by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
+
+(*FIXME DELETE*)
+lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
+by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
+
+lemma NSDERIVD5:
+  "(NSDERIV f x :> D) ==>
+   (\<forall>u. u \<approx> hypreal_of_real x -->
+     ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
+apply (auto simp add: NSDERIV_iff2)
+apply (case_tac "u = hypreal_of_real x", auto)
+apply (drule_tac x = u in spec, auto)
+apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
+apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
+apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
+apply (auto simp add: diff_minus
+	       approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
+		     Infinitesimal_subset_HFinite [THEN subsetD])
+done
+
+lemma NSDERIVD4:
+     "(NSDERIV f x :> D) ==>
+      (\<forall>h \<in> Infinitesimal.
+               (( *f* f)(hypreal_of_real x + h) -
+                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
+apply (auto simp add: nsderiv_def)
+apply (case_tac "h = (0::hypreal) ")
+apply (auto simp add: diff_minus)
+apply (drule_tac x = h in bspec)
+apply (drule_tac [2] c = h in approx_mult1)
+apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
+            simp add: diff_minus)
+done
+
+lemma NSDERIVD3:
+     "(NSDERIV f x :> D) ==>
+      (\<forall>h \<in> Infinitesimal - {0}.
+               (( *f* f)(hypreal_of_real x + h) -
+                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
+apply (auto simp add: nsderiv_def)
+apply (rule ccontr, drule_tac x = h in bspec)
+apply (drule_tac [2] c = h in approx_mult1)
+apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
+            simp add: mult_assoc diff_minus)
+done
+
+text{*Now equivalence between NSDERIV and DERIV*}
+lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
+by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
+
+(*---------------------------------------------------
+         Differentiability implies continuity
+         nice and simple "algebraic" proof
+ --------------------------------------------------*)
+lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
+apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
+apply (drule approx_minus_iff [THEN iffD1])
+apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
+apply (drule_tac x = "-hypreal_of_real x + xa" in bspec)
+ prefer 2 apply (simp add: add_assoc [symmetric]) 
+apply (auto simp add: mem_infmal_iff [symmetric] hypreal_add_commute)
+apply (drule_tac c = "xa + -hypreal_of_real x" in approx_mult1)
+apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
+            simp add: mult_assoc)
+apply (drule_tac x3=D in
+           HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
+             THEN mem_infmal_iff [THEN iffD1]])
+apply (auto simp add: mult_commute 
+            intro: approx_trans approx_minus_iff [THEN iffD2])
+done
+
+text{*Now Sandard proof*}
+lemma DERIV_isCont: "DERIV f x :> D ==> isCont f x"
+by (simp add: NSDERIV_DERIV_iff [symmetric] isNSCont_isCont_iff [symmetric] 
+              NSDERIV_isNSCont)
+
+
+(*----------------------------------------------------------------------------
+      Differentiation rules for combinations of functions
+      follow from clear, straightforard, algebraic
+      manipulations
+ ----------------------------------------------------------------------------*)
+text{*Constant function*}
+
+(* use simple constant nslimit theorem *)
+lemma NSDERIV_const: "(NSDERIV (%x. k) x :> 0)"
+by (simp add: NSDERIV_NSLIM_iff)
+declare NSDERIV_const [simp]
+
+lemma DERIV_const: "(DERIV (%x. k) x :> 0)"
+by (simp add: NSDERIV_DERIV_iff [symmetric])
+declare DERIV_const [simp]
+
+(*-----------------------------------------------------
+    Sum of functions- proved easily
+ ----------------------------------------------------*)
+
+
+lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
+      ==> NSDERIV (%x. f x + g x) x :> Da + Db"
+apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
+apply (auto simp add: add_divide_distrib dest!: spec)
+apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
+apply (auto simp add: add_ac)
+done
+
+(* Standard theorem *)
+lemma DERIV_add: "[| DERIV f x :> Da; DERIV g x :> Db |]
+      ==> DERIV (%x. f x + g x) x :> Da + Db"
+apply (simp add: NSDERIV_add NSDERIV_DERIV_iff [symmetric])
+done
+
+(*-----------------------------------------------------
+  Product of functions - Proof is trivial but tedious
+  and long due to rearrangement of terms
+ ----------------------------------------------------*)
+
+lemma lemma_nsderiv1: "((a::hypreal)*b) + -(c*d) = (b*(a + -c)) + (c*(b + -d))"
+by (simp add: right_distrib)
+
+lemma lemma_nsderiv2: "[| (x + y) / z = hypreal_of_real D + yb; z \<noteq> 0;
+         z \<in> Infinitesimal; yb \<in> Infinitesimal |]
+      ==> x + y \<approx> 0"
+apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
+apply (erule_tac V = " (x + y) / z = hypreal_of_real D + yb" in thin_rl)
+apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
+            simp add: hypreal_mult_assoc mem_infmal_iff [symmetric])
+apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
+done
+
+
+lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
+      ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
+apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
+apply (auto dest!: spec
+	    simp add: starfun_lambda_cancel lemma_nsderiv1)
+apply (simp (no_asm) add: add_divide_distrib)
+apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
+apply (auto simp del: times_divide_eq_right simp add: times_divide_eq_right [symmetric])
+apply (drule_tac D = Db in lemma_nsderiv2)
+apply (drule_tac [4]
+     approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]]) 
+apply (auto intro!: approx_add_mono1 
+            simp add: left_distrib right_distrib mult_commute add_assoc)
+apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)" 
+         in add_commute [THEN subst])
+apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym] 
+                    Infinitesimal_add Infinitesimal_mult 
+                    Infinitesimal_hypreal_of_real_mult 
+                    Infinitesimal_hypreal_of_real_mult2
+          simp add: add_assoc [symmetric])
+done
+
+lemma DERIV_mult:
+     "[| DERIV f x :> Da; DERIV g x :> Db |] 
+      ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
+by (simp add: NSDERIV_mult NSDERIV_DERIV_iff [symmetric])
+
+text{*Multiplying by a constant*}
+lemma NSDERIV_cmult: "NSDERIV f x :> D
+      ==> NSDERIV (%x. c * f x) x :> c*D"
+apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff 
+                  minus_mult_right right_distrib [symmetric])
+apply (erule NSLIM_const [THEN NSLIM_mult])
+done
+
+(* let's do the standard proof though theorem *)
+(* LIM_mult2 follows from a NS proof          *)
+
+lemma DERIV_cmult:
+      "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
+apply (simp only: deriv_def times_divide_eq_right [symmetric] 
+                  NSDERIV_NSLIM_iff minus_mult_right right_distrib [symmetric])
+apply (erule LIM_const [THEN LIM_mult2])
+done
+
+text{*Negation of function*}
+lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
+proof (simp add: NSDERIV_NSLIM_iff)
+  assume "(\<lambda>h. (f (x + h) + - f x) / h) -- 0 --NS> D"
+  hence deriv: "(\<lambda>h. - ((f(x+h) + - f x) / h)) -- 0 --NS> - D" 
+    by (rule NSLIM_minus)
+  have "\<forall>h. - ((f (x + h) + - f x) / h) = (- f (x + h) + f x) / h"
+    by (simp add: minus_divide_left) 
+  with deriv
+  show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
+qed
+
+
+lemma DERIV_minus: "DERIV f x :> D ==> DERIV (%x. -(f x)) x :> -D"
+by (simp add: NSDERIV_minus NSDERIV_DERIV_iff [symmetric])
+
+text{*Subtraction*}
+lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
+by (blast dest: NSDERIV_add NSDERIV_minus)
+
+lemma DERIV_add_minus: "[| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x + -g x) x :> Da + -Db"
+by (blast dest: DERIV_add DERIV_minus)
+
+lemma NSDERIV_diff:
+     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
+      ==> NSDERIV (%x. f x - g x) x :> Da-Db"
+apply (simp add: diff_minus)
+apply (blast intro: NSDERIV_add_minus)
+done
+
+lemma DERIV_diff:
+     "[| DERIV f x :> Da; DERIV g x :> Db |]
+       ==> DERIV (%x. f x - g x) x :> Da-Db"
+apply (simp add: diff_minus)
+apply (blast intro: DERIV_add_minus)
+done
+
+(*---------------------------------------------------------------
+                     (NS) Increment
+ ---------------------------------------------------------------*)
+lemma incrementI:
+      "f NSdifferentiable x ==>
+      increment f x h = ( *f* f) (hypreal_of_real(x) + h) +
+      -hypreal_of_real (f x)"
+by (simp add: increment_def)
+
+lemma incrementI2: "NSDERIV f x :> D ==>
+     increment f x h = ( *f* f) (hypreal_of_real(x) + h) +
+     -hypreal_of_real (f x)"
+apply (erule NSdifferentiableI [THEN incrementI])
+done
+
+(* The Increment theorem -- Keisler p. 65 *)
+lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
+      ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
+apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
+apply (drule bspec, auto)
+apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify) 
+apply (frule_tac b1 = "hypreal_of_real (D) + y" 
+        in hypreal_mult_right_cancel [THEN iffD2])
+apply (erule_tac [2] V = "(( *f* f) (hypreal_of_real (x) + h) + - hypreal_of_real (f x)) / h = hypreal_of_real (D) + y" in thin_rl)
+apply assumption
+apply (simp add: times_divide_eq_right [symmetric] del: times_divide_eq_right)
+apply (auto simp add: left_distrib)
+done
+ 
+lemma increment_thm2:
+     "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
+      ==> \<exists>e \<in> Infinitesimal. increment f x h =
+              hypreal_of_real(D)*h + e*h"
+by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
+
+
+lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
+      ==> increment f x h \<approx> 0"
+apply (drule increment_thm2, 
+       auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
+apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
+done
+
+text{*  Similarly to the above, the chain rule admits an entirely
+   straightforward derivation. Compare this with Harrison's
+   HOL proof of the chain rule, which proved to be trickier and
+   required an alternative characterisation of differentiability-
+   the so-called Carathedory derivative. Our main problem is
+   manipulation of terms.*}
+
+
+(* lemmas *)
+lemma NSDERIV_zero:
+      "[| NSDERIV g x :> D;
+               ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
+               xa \<in> Infinitesimal;
+               xa \<noteq> 0
+            |] ==> D = 0"
+apply (simp add: nsderiv_def)
+apply (drule bspec, auto)
+done
+
+(* can be proved differently using NSLIM_isCont_iff *)
+lemma NSDERIV_approx:
+     "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
+      ==> ( *f* f) (hypreal_of_real(x) + h) + -hypreal_of_real(f x) \<approx> 0"
+apply (simp add: nsderiv_def)
+apply (simp add: mem_infmal_iff [symmetric])
+apply (rule Infinitesimal_ratio)
+apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
+done
+
+(*---------------------------------------------------------------
+   from one version of differentiability
+
+                f(x) - f(a)
+              --------------- \<approx> Db
+                  x - a
+ ---------------------------------------------------------------*)
+lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
+         ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
+         ( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
+      |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
+                   + - hypreal_of_real (f (g x)))
+              / (( *f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real (g x))
+             \<approx> hypreal_of_real(Da)"
+by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
+
+(*--------------------------------------------------------------
+   from other version of differentiability
+
+                f(x + h) - f(x)
+               ----------------- \<approx> Db
+                       h
+ --------------------------------------------------------------*)
+lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
+      ==> (( *f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real(g x)) / xa
+          \<approx> hypreal_of_real(Db)"
+by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
+
+lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
+by auto
+
+(*------------------------------------------------------
+  This proof uses both definitions of differentiability.
+ ------------------------------------------------------*)
+lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
+      ==> NSDERIV (f o g) x :> Da * Db"
+apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
+                mem_infmal_iff [symmetric])
+apply clarify
+apply (frule_tac f = g in NSDERIV_approx)
+apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
+apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
+apply (drule_tac g = g in NSDERIV_zero)
+apply (auto simp add: divide_inverse)
+apply (rule_tac z1 = "( *f* g) (hypreal_of_real (x) + xa) + -hypreal_of_real (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst])
+apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
+apply (rule approx_mult_hypreal_of_real)
+apply (simp_all add: divide_inverse [symmetric])
+apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
+apply (blast intro: NSDERIVD2)
+done
+
+(* standard version *)
+lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
+by (simp add: NSDERIV_DERIV_iff [symmetric] NSDERIV_chain)
+
+lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
+by (auto dest: DERIV_chain simp add: o_def)
+
+text{*Differentiation of natural number powers*}
+lemma NSDERIV_Id: "NSDERIV (%x. x) x :> 1"
+by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def starfun_Id)
+declare NSDERIV_Id [simp]
+
+(*derivative of the identity function*)
+lemma DERIV_Id: "DERIV (%x. x) x :> 1"
+by (simp add: NSDERIV_DERIV_iff [symmetric])
+declare DERIV_Id [simp]
+
+lemmas isCont_Id = DERIV_Id [THEN DERIV_isCont, standard]
+
+(*derivative of linear multiplication*)
+lemma DERIV_cmult_Id: "DERIV (op * c) x :> c"
+by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
+declare DERIV_cmult_Id [simp]
+
+lemma NSDERIV_cmult_Id: "NSDERIV (op * c) x :> c"
+by (simp add: NSDERIV_DERIV_iff)
+declare NSDERIV_cmult_Id [simp]
+
+lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
+apply (induct_tac "n")
+apply (drule_tac [2] DERIV_Id [THEN DERIV_mult])
+apply (auto simp add: real_of_nat_Suc left_distrib)
+apply (case_tac "0 < n")
+apply (drule_tac x = x in realpow_minus_mult)
+apply (auto simp add: real_mult_assoc real_add_commute)
+done
+
+(* NS version *)
+lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
+by (simp add: NSDERIV_DERIV_iff DERIV_pow)
+
+(*---------------------------------------------------------------
+                    Power of -1
+ ---------------------------------------------------------------*)
+
+(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
+lemma NSDERIV_inverse:
+     "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
+apply (simp add: nsderiv_def)
+apply (rule ballI, simp, clarify) 
+apply (frule Infinitesimal_add_not_zero)
+prefer 2 apply (simp add: add_commute) 
+apply (auto simp add: starfun_inverse_inverse realpow_two 
+        simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])
+apply (simp add: inverse_add inverse_mult_distrib [symmetric]
+              inverse_minus_eq [symmetric] add_ac mult_ac
+            del: inverse_mult_distrib inverse_minus_eq 
+                 minus_mult_left [symmetric] minus_mult_right [symmetric])
+apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
+            del: minus_mult_left [symmetric] minus_mult_right [symmetric])
+apply (rule_tac y = " inverse (- hypreal_of_real x * hypreal_of_real x) " in approx_trans)
+apply (rule inverse_add_Infinitesimal_approx2)
+apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal 
+            simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
+apply (rule Infinitesimal_HFinite_mult2, auto)
+done
+
+
+
+
+lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
+by (simp add: NSDERIV_inverse NSDERIV_DERIV_iff [symmetric] del: realpow_Suc)
+
+text{*Derivative of inverse*}
+lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
+      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
+apply (simp only: mult_commute [of d] minus_mult_left power_inverse)
+apply (fold o_def)
+apply (blast intro!: DERIV_chain DERIV_inverse)
+done
+
+lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
+      ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
+by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
+
+text{*Derivative of quotient*}
+lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
+       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) + -(e*f(x))) / (g(x) ^ Suc (Suc 0))"
+apply (drule_tac f = g in DERIV_inverse_fun)
+apply (drule_tac [2] DERIV_mult)
+apply (assumption+)
+apply (simp add: divide_inverse right_distrib power_inverse minus_mult_left
+                 mult_ac 
+     del: realpow_Suc minus_mult_right [symmetric] minus_mult_left [symmetric])
+done
+
+lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
+       ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
+                            + -(e*f(x))) / (g(x) ^ Suc (Suc 0))"
+by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
+
+(* ------------------------------------------------------------------------ *)
+(* Caratheodory formulation of derivative at a point: standard proof        *)
+(* ------------------------------------------------------------------------ *)
+
+lemma CARAT_DERIV:
+     "(DERIV f x :> l) =
+      (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
+      (is "?lhs = ?rhs")
+proof
+  assume der: "DERIV f x :> l"
+  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
+  proof (intro exI conjI)
+    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
+    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
+    show "isCont ?g x" using der 
+      by (simp add: isCont_iff DERIV_iff diff_minus 
+               cong: LIM_equal [rule_format])
+    show "?g x = l" by simp
+  qed
+next
+  assume "?rhs"
+  then obtain g where 
+    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
+  thus "(DERIV f x :> l)" 
+     by (auto simp add: isCont_iff DERIV_iff diff_minus 
+               cong: LIM_equal [rule_format])
+qed
+
+
+lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
+      \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
+by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV)
+
+lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
+by auto
+
+lemma CARAT_DERIVD:
+  assumes all: "\<forall>z. f z - f x = g z * (z-x)"
+      and nsc: "isNSCont g x"
+  shows "NSDERIV f x :> g x"
+proof -
+  from nsc
+  have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
+         ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
+         hypreal_of_real (g x)" 
+    by (simp add: diff_minus isNSCont_def)
+  thus ?thesis using all
+    by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong) 
+qed
+
+(*--------------------------------------------------------------------------*)
+(* Lemmas about nested intervals and proof by bisection (cf.Harrison)       *)
+(* All considerably tidied by lcp                                           *)
+(*--------------------------------------------------------------------------*)
+
+lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"
+apply (induct_tac "no")
+apply (auto intro: order_trans)
+done
+
+lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
+         \<forall>n. g(Suc n) \<le> g(n);
+         \<forall>n. f(n) \<le> g(n) |]
+      ==> Bseq f"
+apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
+apply (induct_tac "n")
+apply (auto intro: order_trans)
+apply (rule_tac y = "g (Suc na) " in order_trans)
+apply (induct_tac [2] "na")
+apply (auto intro: order_trans)
+done
+
+lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
+         \<forall>n. g(Suc n) \<le> g(n);
+         \<forall>n. f(n) \<le> g(n) |]
+      ==> Bseq g"
+apply (subst Bseq_minus_iff [symmetric])
+apply (rule_tac g = "%x. - (f x) " in f_inc_g_dec_Beq_f)
+apply auto
+done
+
+lemma f_inc_imp_le_lim: "[| \<forall>n. f n \<le> f (Suc n);  convergent f |] ==> f n \<le> lim f"
+apply (rule linorder_not_less [THEN iffD1])
+apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
+apply (drule real_less_sum_gt_zero)
+apply (drule_tac x = "f n + - lim f" in spec, safe)
+apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
+apply (subgoal_tac "lim f \<le> f (no + n) ")
+apply (induct_tac [2] "no")
+apply (auto intro: order_trans simp add: diff_minus real_abs_def)
+apply (drule_tac no=no and m=n in lemma_f_mono_add)
+apply (auto simp add: add_commute)
+done
+
+lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
+apply (rule LIMSEQ_minus [THEN limI])
+apply (simp add: convergent_LIMSEQ_iff)
+done
+
+lemma g_dec_imp_lim_le: "[| \<forall>n. g(Suc n) \<le> g(n);  convergent g |] ==> lim g \<le> g n"
+apply (subgoal_tac "- (g n) \<le> - (lim g) ")
+apply (cut_tac [2] f = "%x. - (g x) " in f_inc_imp_le_lim)
+apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
+done
+
+lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
+         \<forall>n. g(Suc n) \<le> g(n);
+         \<forall>n. f(n) \<le> g(n) |]
+      ==> \<exists>l m. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
+                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
+apply (subgoal_tac "monoseq f & monoseq g")
+prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
+apply (subgoal_tac "Bseq f & Bseq g")
+prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
+apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
+apply (rule_tac x = "lim f" in exI)
+apply (rule_tac x = "lim g" in exI)
+apply (auto intro: LIMSEQ_le)
+apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
+done
+
+lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
+         \<forall>n. g(Suc n) \<le> g(n);
+         \<forall>n. f(n) \<le> g(n);
+         (%n. f(n) - g(n)) ----> 0 |]
+      ==> \<exists>l. ((\<forall>n. f(n) \<le> l) & f ----> l) &
+                ((\<forall>n. l \<le> g(n)) & g ----> l)"
+apply (drule lemma_nest, auto)
+apply (subgoal_tac "l = m")
+apply (drule_tac [2] X = f in LIMSEQ_diff)
+apply (auto intro: LIMSEQ_unique)
+done
+
+text{*The universal quantifiers below are required for the declaration
+  of @{text Bolzano_nest_unique} below.*}
+
+lemma Bolzano_bisect_le:
+ "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
+apply (rule allI)
+apply (induct_tac "n")
+apply (auto simp add: Let_def split_def)
+done
+
+lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
+   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
+apply (rule allI)
+apply (induct_tac "n")
+apply (auto simp add: Bolzano_bisect_le Let_def split_def)
+done
+
+lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
+   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
+apply (rule allI)
+apply (induct_tac "n")
+apply (auto simp add: Bolzano_bisect_le Let_def split_def)
+done
+
+lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
+apply auto
+apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
+apply auto
+done
+
+lemma Bolzano_bisect_diff:
+     "a \<le> b ==>
+      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
+      (b-a) / (2 ^ n)"
+apply (induct_tac "n")
+apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
+apply (auto simp add: add_ac Bolzano_bisect_le diff_minus)
+done
+
+lemmas Bolzano_nest_unique =
+    lemma_nest_unique
+    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
+
+
+lemma not_P_Bolzano_bisect:
+  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
+      and notP: "~ P(a,b)"
+      and le:   "a \<le> b"
+  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
+proof (induct n)
+  case 0 thus ?case by simp
+ next
+  case (Suc n)
+  thus ?case
+ by (auto simp del: surjective_pairing [symmetric] 
+             simp add: Let_def split_def Bolzano_bisect_le [OF le] 
+     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"]) 
+qed
+
+(*Now we re-package P_prem as a formula*)
+lemma not_P_Bolzano_bisect':
+     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
+         ~ P(a,b);  a \<le> b |] ==>
+      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
+by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
+
+
+
+lemma lemma_BOLZANO:
+     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
+         \<forall>x. \<exists>d::real. 0 < d &
+                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
+         a \<le> b |]
+      ==> P(a,b)"
+apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
+apply (rule LIMSEQ_minus_cancel)
+apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
+apply (rule ccontr)
+apply (drule not_P_Bolzano_bisect', assumption+)
+apply (rename_tac "l")
+apply (drule_tac x = l in spec, clarify)
+apply (simp add: LIMSEQ_def)
+apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
+apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
+apply (drule real_less_half_sum, auto) 
+apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
+apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
+apply safe
+apply (simp_all (no_asm_simp))
+apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l) " in order_le_less_trans)
+apply (simp (no_asm_simp) add: abs_if)
+apply (rule real_sum_of_halves [THEN subst])
+apply (rule add_strict_mono)
+apply (simp_all add: diff_minus [symmetric])
+done
+
+
+lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
+       (\<forall>x. \<exists>d::real. 0 < d &
+                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
+      --> (\<forall>a b. a \<le> b --> P(a,b))"
+apply clarify
+apply (blast intro: lemma_BOLZANO)
+done
+
+
+subsection{*Intermediate Value Theorem: Prove Contrapositive by Bisection*}
+
+lemma IVT: "[| f(a) \<le> y; y \<le> f(b);
+         a \<le> b;
+         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
+      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
+apply (rule contrapos_pp, assumption)
+apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
+apply safe
+apply simp_all
+apply (simp add: isCont_iff LIM_def)
+apply (rule ccontr)
+apply (subgoal_tac "a \<le> x & x \<le> b")
+ prefer 2
+ apply simp 
+ apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
+apply (drule_tac x = x in spec)+
+apply simp
+apply (drule_tac P = "%r. ?P r --> (\<exists>s. 0<s & ?Q r s) " and x = "\<bar>y - f x\<bar> " in spec)
+apply safe
+apply simp
+apply (drule_tac x = s in spec, clarify)
+apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
+apply (drule_tac x = "ba-x" in spec)
+apply (simp_all add: abs_if)
+apply (drule_tac x = "aa-x" in spec)
+apply (case_tac "x \<le> aa", simp_all)
+apply (drule_tac x = x and y = aa in order_antisym)
+apply (assumption, simp)
+done
+
+lemma IVT2: "[| f(b) \<le> y; y \<le> f(a);
+         a \<le> b;
+         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
+      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
+apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify) 
+apply (drule IVT [where f = "%x. - f x"], assumption)
+apply (auto intro: isCont_minus)
+done
+
+(*HOL style here: object-level formulations*)
+lemma IVT_objl: "(f(a) \<le> y & y \<le> f(b) & a \<le> b &
+      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
+      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
+apply (blast intro: IVT)
+done
+
+lemma IVT2_objl: "(f(b) \<le> y & y \<le> f(a) & a \<le> b &
+      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
+      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
+apply (blast intro: IVT2)
+done
+
+(*---------------------------------------------------------------------------*)
+(* By bisection, function continuous on closed interval is bounded above     *)
+(*---------------------------------------------------------------------------*)
+
+
+lemma isCont_bounded:
+     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
+      ==> \<exists>M. \<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M"
+apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M) " in lemma_BOLZANO2)
+apply safe
+apply simp_all
+apply (rename_tac x xa ya M Ma)
+apply (cut_tac x = M and y = Ma in linorder_linear, safe)
+apply (rule_tac x = Ma in exI, clarify)
+apply (cut_tac x = xb and y = xa in linorder_linear, force)
+apply (rule_tac x = M in exI, clarify)
+apply (cut_tac x = xb and y = xa in linorder_linear, force)
+apply (case_tac "a \<le> x & x \<le> b")
+apply (rule_tac [2] x = 1 in exI)
+prefer 2 apply force
+apply (simp add: LIM_def isCont_iff)
+apply (drule_tac x = x in spec, auto)
+apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
+apply (drule_tac x = 1 in spec, auto)
+apply (rule_tac x = s in exI, clarify)
+apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
+apply (drule_tac x = "xa-x" in spec)
+apply (auto simp add: abs_ge_self, arith+)
+done
+
+(*----------------------------------------------------------------------------*)
+(* Refine the above to existence of least upper bound                         *)
+(*----------------------------------------------------------------------------*)
+
+lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
+      (\<exists>t. isLub UNIV S t)"
+apply (blast intro: reals_complete)
+done
+
+lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
+         ==> \<exists>M. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
+                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
+apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x) " in lemma_reals_complete)
+apply auto
+apply (drule isCont_bounded, assumption)
+apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
+apply (rule exI, auto)
+apply (auto dest!: spec simp add: linorder_not_less) 
+done
+
+(*----------------------------------------------------------------------------*)
+(* Now show that it attains its upper bound                                   *)
+(*----------------------------------------------------------------------------*)
+
+lemma isCont_eq_Ub:
+  assumes le: "a \<le> b"
+      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
+  shows "\<exists>M. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
+             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
+proof -
+  from isCont_has_Ub [OF le con]
+  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
+             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
+  show ?thesis
+  proof (intro exI, intro conjI)
+    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
+    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M" 
+    proof (rule ccontr)
+      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
+      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
+        by (auto simp add: linorder_not_le [symmetric] intro: order_antisym)
+      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
+        by (auto simp add: isCont_inverse isCont_diff con)
+      from isCont_bounded [OF le this]
+      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
+      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
+        by (simp add: M3) 
+      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k 
+        by (auto intro: order_le_less_trans [of _ k]) 
+      with Minv 
+      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))" 
+        by (intro strip less_imp_inverse_less, simp_all)
+      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x" 
+        by simp
+      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le 
+        by (simp, arith)
+      from M2 [OF this]
+      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
+      thus False using invlt [of x] by force
+    qed
+  qed
+qed
+
+
+
+(*----------------------------------------------------------------------------*)
+(* Same theorem for lower bound                                               *)
+(*----------------------------------------------------------------------------*)
+
+lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
+         ==> \<exists>M. (\<forall>x. a \<le> x & x \<le> b --> M \<le> f(x)) &
+                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
+apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
+prefer 2 apply (blast intro: isCont_minus)
+apply (drule_tac f = " (%x. - (f x))" in isCont_eq_Ub)
+apply safe
+apply auto
+done
+
+
+(* ------------------------------------------------------------------------- *)
+(* Another version.                                                          *)
+(* ------------------------------------------------------------------------- *)
+
+lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
+      ==> \<exists>L M. (\<forall>x. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
+          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
+apply (frule isCont_eq_Lb)
+apply (frule_tac [2] isCont_eq_Ub)
+apply (assumption+, safe)
+apply (rule_tac x = "f x" in exI)
+apply (rule_tac x = "f xa" in exI, simp, safe)
+apply (cut_tac x = x and y = xa in linorder_linear, safe)
+apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
+apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
+apply (rule_tac [2] x = xb in exI)
+apply (rule_tac [4] x = xb in exI, simp_all)
+done
+
+(*----------------------------------------------------------------------------*)
+(* If f'(x) > 0 then x is locally strictly increasing at the right            *)
+(*----------------------------------------------------------------------------*)
+
+lemma DERIV_left_inc:
+    "[| DERIV f x :> l;  0 < l |]
+     ==> \<exists>d. 0 < d & (\<forall>h. 0 < h & h < d --> f(x) < f(x + h))"
+apply (simp add: deriv_def LIM_def)
+apply (drule spec, auto)
+apply (rule_tac x = s in exI, auto)
+apply (subgoal_tac "0 < l*h")
+ prefer 2 apply (simp add: zero_less_mult_iff)
+apply (drule_tac x = h in spec)
+apply (simp add: real_abs_def pos_le_divide_eq pos_less_divide_eq 
+            split add: split_if_asm)
+done
+
+lemma DERIV_left_dec:
+  assumes der: "DERIV f x :> l"
+      and l:   "l < 0"
+  shows "\<exists>d. 0 < d & (\<forall>h. 0 < h & h < d --> f(x) < f(x-h))"
+proof -
+  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
+  have "\<exists>s. 0 < s \<and>
+              (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
+    by (simp add: diff_minus)
+  then obtain s
+        where s:   "0 < s" 
+          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
+    by auto
+  thus ?thesis
+  proof (intro exI conjI strip)
+    show "0<s" .
+    fix h::real
+    assume "0 < h \<and> h < s"
+    with all [of "-h"] show "f x < f (x-h)" 
+    proof (simp add: real_abs_def pos_less_divide_eq diff_minus [symmetric] 
+		split add: split_if_asm)
+      assume "~ l \<le> - ((f (x-h) - f x) / h)" and h: "0 < h" 
+      with l 
+      have "0 < (f (x-h) - f x) / h" by arith
+      thus "f x < f (x-h)"
+	by (simp add: pos_less_divide_eq h)
+    qed
+  qed
+qed
+
+lemma DERIV_local_max: 
+  assumes der: "DERIV f x :> l"
+      and d:   "0 < d"
+      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
+  shows "l = 0"
+proof (cases rule: linorder_cases [of l 0])
+  case equal show ?thesis .
+next
+  case less
+  from DERIV_left_dec [OF der less]
+  obtain d' where d': "0 < d'"
+             and lt: "\<forall>h. 0 < h \<and> h < d' \<longrightarrow> f x < f (x-h)" by blast
+  from real_lbound_gt_zero [OF d d']
+  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
+  with lt le [THEN spec [where x="x-e"]] 
+  show ?thesis by (auto simp add: abs_if)
+next
+  case greater
+  from DERIV_left_inc [OF der greater]
+  obtain d' where d': "0 < d'"
+             and lt: "\<forall>h. 0 < h \<and> h < d' \<longrightarrow> f x < f (x + h)" by blast
+  from real_lbound_gt_zero [OF d d']
+  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
+  with lt le [THEN spec [where x="x+e"]]
+  show ?thesis by (auto simp add: abs_if)
+qed
+
+
+text{*Similar theorem for a local minimum*}
+lemma DERIV_local_min:
+     "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
+by (drule DERIV_minus [THEN DERIV_local_max], auto)
+
+
+text{*In particular, if a function is locally flat*}
+lemma DERIV_local_const:
+     "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
+by (auto dest!: DERIV_local_max)
+
+text{*Lemma about introducing open ball in open interval*}
+lemma lemma_interval_lt:
+     "[| a < x;  x < b |] 
+      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
+apply (simp add: abs_interval_iff)
+apply (insert linorder_linear [of "x-a" "b-x"], safe)
+apply (rule_tac x = "x-a" in exI)
+apply (rule_tac [2] x = "b-x" in exI, auto)
+done
+
+lemma lemma_interval: "[| a < x;  x < b |] ==>
+        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
+apply (drule lemma_interval_lt, auto)
+apply (auto intro!: exI)
+done
+
+text{*Rolle's Theorem.
+   If @{term f} is defined and continuous on the closed interval 
+   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"}, 
+   and @{term "f(a) = f(b)"},
+   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
+theorem Rolle: 
+  assumes lt: "a < b"
+      and eq: "f(a) = f(b)"
+      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
+      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
+  shows "\<exists>z. a < z & z < b & DERIV f z :> 0"
+proof -
+  have le: "a \<le> b" using lt by simp
+  from isCont_eq_Ub [OF le con]
+  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x" 
+             and alex: "a \<le> x" and xleb: "x \<le> b" 
+    by blast
+  from isCont_eq_Lb [OF le con]
+  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z" 
+              and alex': "a \<le> x'" and x'leb: "x' \<le> b" 
+    by blast
+  show ?thesis
+  proof cases
+    assume axb: "a < x & x < b"
+        --{*@{term f} attains its maximum within the interval*}
+    hence ax: "a<x" and xb: "x<b" by auto
+    from lemma_interval [OF ax xb]
+    obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
+      by blast
+    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
+      by blast
+    from differentiableD [OF dif [OF axb]]
+    obtain l where der: "DERIV f x :> l" ..
+    have "l=0" by (rule DERIV_local_max [OF der d bound']) 
+        --{*the derivative at a local maximum is zero*}
+    thus ?thesis using ax xb der by auto
+  next
+    assume notaxb: "~ (a < x & x < b)"
+    hence xeqab: "x=a | x=b" using alex xleb by arith
+    hence fb_eq_fx: "f b = f x" by (auto simp add: eq) 
+    show ?thesis
+    proof cases
+      assume ax'b: "a < x' & x' < b"
+        --{*@{term f} attains its minimum within the interval*}
+      hence ax': "a<x'" and x'b: "x'<b" by auto
+      from lemma_interval [OF ax' x'b]
+      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
+	by blast
+      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
+	by blast
+      from differentiableD [OF dif [OF ax'b]]
+      obtain l where der: "DERIV f x' :> l" ..
+      have "l=0" by (rule DERIV_local_min [OF der d bound']) 
+        --{*the derivative at a local minimum is zero*}
+      thus ?thesis using ax' x'b der by auto
+    next
+      assume notax'b: "~ (a < x' & x' < b)"
+        --{*@{term f} is constant througout the interval*}
+      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
+      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq) 
+      from dense [OF lt]
+      obtain r where ar: "a < r" and rb: "r < b" by blast
+      from lemma_interval [OF ar rb]
+      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
+	by blast
+      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b" 
+      proof (clarify) 
+        fix z::real
+        assume az: "a \<le> z" and zb: "z \<le> b"
+        show "f z = f b"
+        proof (rule order_antisym)
+          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb) 
+          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb) 
+        qed
+      qed
+      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
+      proof (intro strip)
+        fix y::real
+        assume lt: "\<bar>r-y\<bar> < d"
+        hence "f y = f b" by (simp add: eq_fb bound) 
+        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
+      qed
+      from differentiableD [OF dif [OF conjI [OF ar rb]]]
+      obtain l where der: "DERIV f r :> l" ..
+      have "l=0" by (rule DERIV_local_const [OF der d bound']) 
+        --{*the derivative of a constant function is zero*}
+      thus ?thesis using ar rb der by auto
+    qed
+  qed
+qed
+
+
+subsection{*Mean Value Theorem*}
+
+lemma lemma_MVT:
+     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
+proof cases
+  assume "a=b" thus ?thesis by simp
+next
+  assume "a\<noteq>b" 
+  hence ba: "b-a \<noteq> 0" by arith
+  show ?thesis
+    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
+        simp add: right_diff_distrib, simp add: left_diff_distrib)
+qed
+
+theorem MVT: 
+  assumes lt:  "a < b"
+      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
+      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
+  shows "\<exists>l z. a < z & z < b & DERIV f z :> l &
+                   (f(b) - f(a) = (b-a) * l)"
+proof -
+  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
+  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
+    by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id) 
+  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
+  proof (clarify)
+    fix x::real
+    assume ax: "a < x" and xb: "x < b"
+    from differentiableD [OF dif [OF conjI [OF ax xb]]]
+    obtain l where der: "DERIV f x :> l" ..
+    show "?F differentiable x"
+      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
+          blast intro: DERIV_diff DERIV_cmult_Id der) 
+  qed  
+  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
+  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0" 
+    by blast
+  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
+    by (rule DERIV_cmult_Id)
+  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z 
+                   :> 0 + (f b - f a) / (b - a)"
+    by (rule DERIV_add [OF der])
+  show ?thesis  
+  proof (intro exI conjI)
+    show "a < z" .
+    show "z < b" .
+    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by simp
+    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
+  qed
+qed
+
+
+text{*A function is constant if its derivative is 0 over an interval.*}
+
+lemma DERIV_isconst_end: "[| a < b;
+         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
+         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
+        ==> (f b = f a)"
+apply (drule MVT, assumption)
+apply (blast intro: differentiableI)
+apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
+done
+
+lemma DERIV_isconst1: "[| a < b;
+         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
+         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
+        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
+apply safe
+apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
+apply (drule_tac b = x in DERIV_isconst_end, auto)
+done
+
+lemma DERIV_isconst2: "[| a < b;
+         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
+         \<forall>x. a < x & x < b --> DERIV f x :> 0;
+         a \<le> x; x \<le> b |]
+        ==> f x = f a"
+apply (blast dest: DERIV_isconst1)
+done
+
+lemma DERIV_isconst_all: "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
+apply (rule linorder_cases [of x y])
+apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
+done
+
+lemma DERIV_const_ratio_const:
+     "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
+apply (rule linorder_cases [of a b], auto)
+apply (drule_tac [!] f = f in MVT)
+apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
+apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
+done
+
+lemma DERIV_const_ratio_const2:
+     "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
+apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
+apply (auto dest!: DERIV_const_ratio_const simp add: real_mult_assoc)
+done
+
+lemma real_average_minus_first: "((a + b) /2 - a) = (b-a)/(2::real)"
+by auto
+declare real_average_minus_first [simp]
+
+lemma real_average_minus_second: "((b + a)/2 - a) = (b-a)/(2::real)"
+by auto
+declare real_average_minus_second [simp]
+
+text{*Gallileo's "trick": average velocity = av. of end velocities*}
+
+lemma DERIV_const_average:
+  assumes neq: "a \<noteq> (b::real)"
+      and der: "\<forall>x. DERIV v x :> k"
+  shows "v ((a + b)/2) = (v a + v b)/2"
+proof (cases rule: linorder_cases [of a b])
+  case equal with neq show ?thesis by simp
+next
+  case less
+  have "(v b - v a) / (b - a) = k"
+    by (rule DERIV_const_ratio_const2 [OF neq der])
+  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp 
+  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
+    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
+  ultimately show ?thesis using neq by force
+next
+  case greater
+  have "(v b - v a) / (b - a) = k"
+    by (rule DERIV_const_ratio_const2 [OF neq der])
+  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp 
+  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
+    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
+  ultimately show ?thesis using neq by (force simp add: add_commute) 
+qed
+
+
+text{*Dull lemma: an continuous injection on an interval must have a
+strict maximum at an end point, not in the middle.*}
+
+lemma lemma_isCont_inj:
+  assumes d: "0 < d"
+      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
+      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
+  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
+proof (rule ccontr)
+  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
+  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto 
+  show False
+  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
+    case le
+    from d cont all [of "x+d"]
+    have flef: "f(x+d) \<le> f x" 
+     and xlex: "x - d \<le> x" 
+     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z" 
+       by (auto simp add: abs_if)
+    from IVT [OF le flef xlex cont']
+    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
+    moreover
+    hence "g(f x') = g (f(x+d))" by simp
+    ultimately show False using d inj [of x'] inj [of "x+d"]
+      by (simp add: abs_le_interval_iff)
+  next
+    case ge
+    from d cont all [of "x-d"]
+    have flef: "f(x-d) \<le> f x" 
+     and xlex: "x \<le> x+d" 
+     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" 
+       by (auto simp add: abs_if)
+    from IVT2 [OF ge flef xlex cont']
+    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
+    moreover
+    hence "g(f x') = g (f(x-d))" by simp
+    ultimately show False using d inj [of x'] inj [of "x-d"]
+      by (simp add: abs_le_interval_iff)
+  qed
+qed
+
+
+text{*Similar version for lower bound.*}
+
+lemma lemma_isCont_inj2:
+     "[|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>z. \<bar>z-x\<bar> \<le> d & f z < f x"
+apply (insert lemma_isCont_inj
+          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
+apply (simp add: isCont_minus linorder_not_le) 
+done
+
+text{*Show there's an interval surrounding @{term "f(x)"} in 
+@{text "f[[x - d, x + d]]"} .*}
+
+lemma isCont_inj_range: 
+  assumes d: "0 < d"
+      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
+      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
+  shows "\<exists>e. 0<e & (\<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y))"
+proof -
+  have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
+    by (auto simp add: abs_le_interval_iff)
+  from isCont_Lb_Ub [OF this]
+  obtain L M 
+  where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
+    and all2 [rule_format]:
+           "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
+    by auto
+  with d have "L \<le> f x & f x \<le> M" by simp
+  moreover have "L \<noteq> f x"
+  proof -
+    from lemma_isCont_inj2 [OF d inj cont]
+    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
+    thus ?thesis using all1 [of u] by arith
+  qed
+  moreover have "f x \<noteq> M"
+  proof -
+    from lemma_isCont_inj [OF d inj cont]
+    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
+    thus ?thesis using all1 [of u] by arith
+  qed
+  ultimately have "L < f x & f x < M" by arith
+  hence "0 < f x - L" "0 < M - f x" by arith+
+  from real_lbound_gt_zero [OF this]
+  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
+  thus ?thesis
+  proof (intro exI conjI)
+    show "0<e" .
+    show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
+    proof (intro strip)
+      fix y::real
+      assume "\<bar>y - f x\<bar> \<le> e"
+      with e have "L \<le> y \<and> y \<le> M" by arith
+      from all2 [OF this]
+      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
+      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y" 
+        by (force simp add: abs_le_interval_iff)
+    qed
+  qed
+qed
+
+
+text{*Continuity of inverse function*}
+
+lemma isCont_inverse_function:
+  assumes d: "0 < d"
+      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
+      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
+  shows "isCont g (f x)"
+proof (simp add: isCont_iff LIM_eq)
+  show "\<forall>r. 0 < r \<longrightarrow>
+         (\<exists>s. 0<s \<and> (\<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r))"
+  proof (intro strip)
+    fix r::real
+    assume r: "0<r"
+    from real_lbound_gt_zero [OF r d]
+    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
+    with inj cont
+    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z" 
+                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
+    from isCont_inj_range [OF e this]
+    obtain e' where e': "0 < e'" 
+        and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
+          by blast
+    show "\<exists>s. 0<s \<and> (\<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
+    proof (intro exI conjI)
+      show "0<e'" .
+      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
+      proof (intro strip)
+        fix z::real
+        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
+        with e e_lt e_simps all [rule_format, of "f x + z"]
+        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
+      qed
+    qed
+  qed
+qed  
+
+ML
+{*
+val LIM_def = thm"LIM_def";
+val NSLIM_def = thm"NSLIM_def";
+val isCont_def = thm"isCont_def";
+val isNSCont_def = thm"isNSCont_def";
+val deriv_def = thm"deriv_def";
+val nsderiv_def = thm"nsderiv_def";
+val differentiable_def = thm"differentiable_def";
+val NSdifferentiable_def = thm"NSdifferentiable_def";
+val increment_def = thm"increment_def";
+val isUCont_def = thm"isUCont_def";
+val isNSUCont_def = thm"isNSUCont_def";
+
+val half_gt_zero_iff = thm "half_gt_zero_iff";
+val half_gt_zero = thms "half_gt_zero";
+val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq";
+val LIM_eq = thm "LIM_eq";
+val LIM_D = thm "LIM_D";
+val LIM_const = thm "LIM_const";
+val LIM_add = thm "LIM_add";
+val LIM_minus = thm "LIM_minus";
+val LIM_add_minus = thm "LIM_add_minus";
+val LIM_diff = thm "LIM_diff";
+val LIM_const_not_eq = thm "LIM_const_not_eq";
+val LIM_const_eq = thm "LIM_const_eq";
+val LIM_unique = thm "LIM_unique";
+val LIM_mult_zero = thm "LIM_mult_zero";
+val LIM_self = thm "LIM_self";
+val LIM_equal = thm "LIM_equal";
+val LIM_trans = thm "LIM_trans";
+val LIM_NSLIM = thm "LIM_NSLIM";
+val NSLIM_LIM = thm "NSLIM_LIM";
+val LIM_NSLIM_iff = thm "LIM_NSLIM_iff";
+val NSLIM_mult = thm "NSLIM_mult";
+val LIM_mult2 = thm "LIM_mult2";
+val NSLIM_add = thm "NSLIM_add";
+val LIM_add2 = thm "LIM_add2";
+val NSLIM_const = thm "NSLIM_const";
+val LIM_const2 = thm "LIM_const2";
+val NSLIM_minus = thm "NSLIM_minus";
+val LIM_minus2 = thm "LIM_minus2";
+val NSLIM_add_minus = thm "NSLIM_add_minus";
+val LIM_add_minus2 = thm "LIM_add_minus2";
+val NSLIM_inverse = thm "NSLIM_inverse";
+val LIM_inverse = thm "LIM_inverse";
+val NSLIM_zero = thm "NSLIM_zero";
+val LIM_zero2 = thm "LIM_zero2";
+val NSLIM_zero_cancel = thm "NSLIM_zero_cancel";
+val LIM_zero_cancel = thm "LIM_zero_cancel";
+val NSLIM_not_zero = thm "NSLIM_not_zero";
+val NSLIM_const_not_eq = thm "NSLIM_const_not_eq";
+val NSLIM_const_eq = thm "NSLIM_const_eq";
+val NSLIM_unique = thm "NSLIM_unique";
+val LIM_unique2 = thm "LIM_unique2";
+val NSLIM_mult_zero = thm "NSLIM_mult_zero";
+val LIM_mult_zero2 = thm "LIM_mult_zero2";
+val NSLIM_self = thm "NSLIM_self";
+val isNSContD = thm "isNSContD";
+val isNSCont_NSLIM = thm "isNSCont_NSLIM";
+val NSLIM_isNSCont = thm "NSLIM_isNSCont";
+val isNSCont_NSLIM_iff = thm "isNSCont_NSLIM_iff";
+val isNSCont_LIM_iff = thm "isNSCont_LIM_iff";
+val isNSCont_isCont_iff = thm "isNSCont_isCont_iff";
+val isCont_isNSCont = thm "isCont_isNSCont";
+val isNSCont_isCont = thm "isNSCont_isCont";
+val NSLIM_h_iff = thm "NSLIM_h_iff";
+val NSLIM_isCont_iff = thm "NSLIM_isCont_iff";
+val LIM_isCont_iff = thm "LIM_isCont_iff";
+val isCont_iff = thm "isCont_iff";
+val isCont_add = thm "isCont_add";
+val isCont_mult = thm "isCont_mult";
+val isCont_o = thm "isCont_o";
+val isCont_o2 = thm "isCont_o2";
+val isNSCont_minus = thm "isNSCont_minus";
+val isCont_minus = thm "isCont_minus";
+val isCont_inverse = thm "isCont_inverse";
+val isNSCont_inverse = thm "isNSCont_inverse";
+val isCont_diff = thm "isCont_diff";
+val isCont_const = thm "isCont_const";
+val isNSCont_const = thm "isNSCont_const";
+val isNSCont_rabs = thm "isNSCont_rabs";
+val isCont_rabs = thm "isCont_rabs";
+val isNSUContD = thm "isNSUContD";
+val isUCont_isCont = thm "isUCont_isCont";
+val isUCont_isNSUCont = thm "isUCont_isNSUCont";
+val isNSUCont_isUCont = thm "isNSUCont_isUCont";
+val DERIV_iff = thm "DERIV_iff";
+val DERIV_NS_iff = thm "DERIV_NS_iff";
+val DERIV_D = thm "DERIV_D";
+val NS_DERIV_D = thm "NS_DERIV_D";
+val DERIV_unique = thm "DERIV_unique";
+val NSDeriv_unique = thm "NSDeriv_unique";
+val differentiableD = thm "differentiableD";
+val differentiableI = thm "differentiableI";
+val NSdifferentiableD = thm "NSdifferentiableD";
+val NSdifferentiableI = thm "NSdifferentiableI";
+val LIM_I = thm "LIM_I";
+val DERIV_LIM_iff = thm "DERIV_LIM_iff";
+val DERIV_iff2 = thm "DERIV_iff2";
+val NSDERIV_NSLIM_iff = thm "NSDERIV_NSLIM_iff";
+val NSDERIV_NSLIM_iff2 = thm "NSDERIV_NSLIM_iff2";
+val NSDERIV_iff2 = thm "NSDERIV_iff2";
+val hypreal_not_eq_minus_iff = thm "hypreal_not_eq_minus_iff";
+val NSDERIVD5 = thm "NSDERIVD5";
+val NSDERIVD4 = thm "NSDERIVD4";
+val NSDERIVD3 = thm "NSDERIVD3";
+val NSDERIV_DERIV_iff = thm "NSDERIV_DERIV_iff";
+val NSDERIV_isNSCont = thm "NSDERIV_isNSCont";
+val DERIV_isCont = thm "DERIV_isCont";
+val NSDERIV_const = thm "NSDERIV_const";
+val DERIV_const = thm "DERIV_const";
+val NSDERIV_add = thm "NSDERIV_add";
+val DERIV_add = thm "DERIV_add";
+val NSDERIV_mult = thm "NSDERIV_mult";
+val DERIV_mult = thm "DERIV_mult";
+val NSDERIV_cmult = thm "NSDERIV_cmult";
+val DERIV_cmult = thm "DERIV_cmult";
+val NSDERIV_minus = thm "NSDERIV_minus";
+val DERIV_minus = thm "DERIV_minus";
+val NSDERIV_add_minus = thm "NSDERIV_add_minus";
+val DERIV_add_minus = thm "DERIV_add_minus";
+val NSDERIV_diff = thm "NSDERIV_diff";
+val DERIV_diff = thm "DERIV_diff";
+val incrementI = thm "incrementI";
+val incrementI2 = thm "incrementI2";
+val increment_thm = thm "increment_thm";
+val increment_thm2 = thm "increment_thm2";
+val increment_approx_zero = thm "increment_approx_zero";
+val NSDERIV_zero = thm "NSDERIV_zero";
+val NSDERIV_approx = thm "NSDERIV_approx";
+val NSDERIVD1 = thm "NSDERIVD1";
+val NSDERIVD2 = thm "NSDERIVD2";
+val NSDERIV_chain = thm "NSDERIV_chain";
+val DERIV_chain = thm "DERIV_chain";
+val DERIV_chain2 = thm "DERIV_chain2";
+val NSDERIV_Id = thm "NSDERIV_Id";
+val DERIV_Id = thm "DERIV_Id";
+val isCont_Id = thms "isCont_Id";
+val DERIV_cmult_Id = thm "DERIV_cmult_Id";
+val NSDERIV_cmult_Id = thm "NSDERIV_cmult_Id";
+val DERIV_pow = thm "DERIV_pow";
+val NSDERIV_pow = thm "NSDERIV_pow";
+val NSDERIV_inverse = thm "NSDERIV_inverse";
+val DERIV_inverse = thm "DERIV_inverse";
+val DERIV_inverse_fun = thm "DERIV_inverse_fun";
+val NSDERIV_inverse_fun = thm "NSDERIV_inverse_fun";
+val DERIV_quotient = thm "DERIV_quotient";
+val NSDERIV_quotient = thm "NSDERIV_quotient";
+val CARAT_DERIV = thm "CARAT_DERIV";
+val CARAT_NSDERIV = thm "CARAT_NSDERIV";
+val hypreal_eq_minus_iff3 = thm "hypreal_eq_minus_iff3";
+val starfun_if_eq = thm "starfun_if_eq";
+val CARAT_DERIVD = thm "CARAT_DERIVD";
+val f_inc_g_dec_Beq_f = thm "f_inc_g_dec_Beq_f";
+val f_inc_g_dec_Beq_g = thm "f_inc_g_dec_Beq_g";
+val f_inc_imp_le_lim = thm "f_inc_imp_le_lim";
+val lim_uminus = thm "lim_uminus";
+val g_dec_imp_lim_le = thm "g_dec_imp_lim_le";
+val Bolzano_bisect_le = thm "Bolzano_bisect_le";
+val Bolzano_bisect_fst_le_Suc = thm "Bolzano_bisect_fst_le_Suc";
+val Bolzano_bisect_Suc_le_snd = thm "Bolzano_bisect_Suc_le_snd";
+val eq_divide_2_times_iff = thm "eq_divide_2_times_iff";
+val Bolzano_bisect_diff = thm "Bolzano_bisect_diff";
+val Bolzano_nest_unique = thms "Bolzano_nest_unique";
+val not_P_Bolzano_bisect = thm "not_P_Bolzano_bisect";
+val not_P_Bolzano_bisect = thm "not_P_Bolzano_bisect";
+val lemma_BOLZANO2 = thm "lemma_BOLZANO2";
+val IVT = thm "IVT";
+val IVT2 = thm "IVT2";
+val IVT_objl = thm "IVT_objl";
+val IVT2_objl = thm "IVT2_objl";
+val isCont_bounded = thm "isCont_bounded";
+val isCont_has_Ub = thm "isCont_has_Ub";
+val isCont_eq_Ub = thm "isCont_eq_Ub";
+val isCont_eq_Lb = thm "isCont_eq_Lb";
+val isCont_Lb_Ub = thm "isCont_Lb_Ub";
+val DERIV_left_inc = thm "DERIV_left_inc";
+val DERIV_left_dec = thm "DERIV_left_dec";
+val DERIV_local_max = thm "DERIV_local_max";
+val DERIV_local_min = thm "DERIV_local_min";
+val DERIV_local_const = thm "DERIV_local_const";
+val Rolle = thm "Rolle";
+val MVT = thm "MVT";
+val DERIV_isconst_end = thm "DERIV_isconst_end";
+val DERIV_isconst1 = thm "DERIV_isconst1";
+val DERIV_isconst2 = thm "DERIV_isconst2";
+val DERIV_isconst_all = thm "DERIV_isconst_all";
+val DERIV_const_ratio_const = thm "DERIV_const_ratio_const";
+val DERIV_const_ratio_const2 = thm "DERIV_const_ratio_const2";
+val real_average_minus_first = thm "real_average_minus_first";
+val real_average_minus_second = thm "real_average_minus_second";
+val DERIV_const_average = thm "DERIV_const_average";
+val isCont_inj_range = thm "isCont_inj_range";
+val isCont_inverse_function = thm "isCont_inverse_function";
+*}
+
 
 end
 
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/NSA.thy
--- a/src/HOL/Hyperreal/NSA.thy	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/NSA.thy	Fri Mar 19 10:51:03 2004 +0100
@@ -154,7 +154,7 @@
 lemma SReal_dense:
      "[| (x::hypreal) \<in> Reals; y \<in> Reals;  x<y |] ==> \<exists>r \<in> Reals. x<r & r<y"
 apply (auto simp add: SReal_iff)
-apply (drule real_dense, safe)
+apply (drule dense, safe)
 apply (rule_tac x = "hypreal_of_real r" in bexI, auto)
 done
 
@@ -304,14 +304,11 @@
 lemma hypreal_sum_of_halves: "x/(2::hypreal) + x/(2::hypreal) = x"
 by auto
 
-lemma hypreal_half_gt_zero: "0 < r ==> 0 < r/(2::hypreal)"
-by auto
-
 lemma Infinitesimal_add:
      "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> (x+y) \<in> Infinitesimal"
 apply (auto simp add: Infinitesimal_def)
 apply (rule hypreal_sum_of_halves [THEN subst])
-apply (drule hypreal_half_gt_zero)
+apply (drule half_gt_zero)
 apply (blast intro: hrabs_add_less hrabs_add_less SReal_divide_number_of)
 done
 
@@ -488,6 +485,9 @@
 by (simp add: approx_def Infinitesimal_def)
 declare approx_refl [iff]
 
+lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
+by (simp add: hypreal_add_commute)
+
 lemma approx_sym: "x @= y ==> y @= x"
 apply (simp add: approx_def)
 apply (rule hypreal_minus_distrib1 [THEN subst])
@@ -562,7 +562,6 @@
 val InfinitesimalD = thm "InfinitesimalD";
 val Infinitesimal_zero = thm "Infinitesimal_zero";
 val hypreal_sum_of_halves = thm "hypreal_sum_of_halves";
-val hypreal_half_gt_zero = thm "hypreal_half_gt_zero";
 val Infinitesimal_add = thm "Infinitesimal_add";
 val Infinitesimal_minus_iff = thm "Infinitesimal_minus_iff";
 val Infinitesimal_diff = thm "Infinitesimal_diff";
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/NthRoot.thy
--- a/src/HOL/Hyperreal/NthRoot.thy	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/NthRoot.thy	Fri Mar 19 10:51:03 2004 +0100
@@ -1,8 +1,7 @@
 (*  Title       : NthRoot.thy
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
-    Description : Existence of nth root. Adapted from
-                   http://www.math.unl.edu/~webnotes
+    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
 *)
 
 header{*Existence of Nth Root*}
@@ -17,10 +16,9 @@
 lemma lemma_nth_realpow_non_empty:
      "[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"
 apply (case_tac "1 <= a")
-apply (rule_tac x = "1" in exI)
+apply (rule_tac x = 1 in exI)
 apply (drule_tac [2] linorder_not_le [THEN iffD1])
-apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc])
-apply (simp add: ); 
+apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp) 
 apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc)
 done
 
@@ -32,23 +30,19 @@
      "[| (0::real) < a; 0 < n |]  
       ==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"
 apply (case_tac "1 <= a")
-apply (rule_tac x = "a" in exI)
+apply (rule_tac x = a in exI)
 apply (drule_tac [2] linorder_not_le [THEN iffD1])
-apply (rule_tac [2] x = "1" in exI)
-apply (rule_tac [!] setleI [THEN isUbI])
-apply safe
+apply (rule_tac [2] x = 1 in exI)
+apply (rule_tac [!] setleI [THEN isUbI], safe)
 apply (simp_all (no_asm))
 apply (rule_tac [!] ccontr)
 apply (drule_tac [!] linorder_not_le [THEN iffD1])
-apply (drule realpow_ge_self2 , assumption)
-apply (drule_tac n = "n" in realpow_less)
+apply (drule realpow_ge_self2, assumption)
+apply (drule_tac n = n in realpow_less)
 apply (assumption+)
-apply (drule real_le_trans , assumption)
-apply (drule_tac y = "y ^ n" in order_less_le_trans)
-apply (assumption)
-apply (simp); 
-apply (drule_tac n = "n" in zero_less_one [THEN realpow_less])
-apply auto
+apply (drule real_le_trans, assumption)
+apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp) 
+apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto)
 done
 
 lemma nth_realpow_isLub_ex:
@@ -62,23 +56,23 @@
 lemma lemma_nth_realpow_seq:
      "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u  
            ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"
-apply (safe , drule isLubD2 , blast)
+apply (safe, drule isLubD2, blast)
 apply (simp add: linorder_not_less [symmetric])
 done
 
 lemma lemma_nth_realpow_isLub_gt_zero:
      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
          0 < a; 0 < n |] ==> 0 < u"
-apply (drule lemma_nth_realpow_non_empty , auto)
-apply (drule_tac y = "s" in isLub_isUb [THEN isUbD])
+apply (drule lemma_nth_realpow_non_empty, auto)
+apply (drule_tac y = s in isLub_isUb [THEN isUbD])
 apply (auto intro: order_less_le_trans)
 done
 
 lemma lemma_nth_realpow_isLub_ge:
      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
          0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"
-apply (safe)
-apply (frule lemma_nth_realpow_seq , safe)
+apply safe
+apply (frule lemma_nth_realpow_seq, safe)
 apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric])
 apply (simp add: linorder_not_less)
 apply (rule order_less_trans [of _ 0])
@@ -90,7 +84,7 @@
      "[| (0::real) < a; 0 < n;  
      isLub (UNIV::real set)  
      {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"
-apply (frule lemma_nth_realpow_isLub_ge , safe)
+apply (frule lemma_nth_realpow_isLub_ge, safe)
 apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])
 apply (auto simp add: real_of_nat_def)
 done
@@ -100,16 +94,14 @@
 lemma less_isLub_not_isUb:
      "[| isLub (UNIV::real set) S u; x < u |]  
            ==> ~ isUb (UNIV::real set) S x"
-apply (safe)
-apply (drule isLub_le_isUb)
-apply assumption
-apply (drule order_less_le_trans)
-apply (auto)
+apply safe
+apply (drule isLub_le_isUb, assumption)
+apply (drule order_less_le_trans, auto)
 done
 
 lemma not_isUb_less_ex:
      "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"
-apply (rule ccontr , erule swap)
+apply (rule ccontr, erule swap)
 apply (rule setleI [THEN isUbI])
 apply (auto simp add: linorder_not_less [symmetric])
 done
@@ -129,14 +121,12 @@
 lemma lemma_nth_realpow_isLub_le:
      "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  
        0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a"
-apply (safe)
+apply safe
 apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])
-apply (rule_tac n = "k" in real_mult_less_self)
-apply (blast intro: lemma_nth_realpow_isLub_gt_zero)
-apply (safe)
-apply (drule_tac n = "k" in
-        lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero])
-apply assumption+
+apply (rule_tac n = k in real_mult_less_self)
+apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe)
+apply (drule_tac n = k in
+        lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+)
 apply (blast intro: order_trans order_less_imp_le power_mono) 
 done
 
@@ -145,7 +135,7 @@
      "[| (0::real) < a; 0 < n;  
      isLub (UNIV::real set)  
      {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"
-apply (frule lemma_nth_realpow_isLub_le , safe)
+apply (frule lemma_nth_realpow_isLub_le, safe)
 apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult
                 [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])
 apply (auto simp add: real_of_nat_def)
@@ -153,29 +143,26 @@
 
 text{*The theorem at last!*}
 lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"
-apply (frule nth_realpow_isLub_ex , auto)
-apply (auto intro: realpow_nth_le realpow_nth_ge real_le_anti_sym)
+apply (frule nth_realpow_isLub_ex, auto)
+apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym)
 done
 
 (* positive only *)
 lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"
-apply (frule nth_realpow_isLub_ex , auto)
-apply (auto intro: realpow_nth_le realpow_nth_ge real_le_anti_sym lemma_nth_realpow_isLub_gt_zero)
+apply (frule nth_realpow_isLub_ex, auto)
+apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero)
 done
 
 lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"
-apply (blast intro: realpow_pos_nth)
-done
+by (blast intro: realpow_pos_nth)
 
 (* uniqueness of nth positive root *)
 lemma realpow_pos_nth_unique:
      "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"
 apply (auto intro!: realpow_pos_nth)
-apply (cut_tac x = "r" and y = "y" in linorder_less_linear)
-apply auto
-apply (drule_tac x = "r" in realpow_less)
-apply (drule_tac [4] x = "y" in realpow_less)
-apply (auto)
+apply (cut_tac x = r and y = y in linorder_less_linear, auto)
+apply (drule_tac x = r in realpow_less)
+apply (drule_tac [4] x = y in realpow_less, auto)
 done
 
 ML
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/SEQ.ML
--- a/src/HOL/Hyperreal/SEQ.ML	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/SEQ.ML	Fri Mar 19 10:51:03 2004 +0100
@@ -618,7 +618,7 @@
 \              |] ==> ALL n. ma <= n --> X n = X ma";
 by (Step_tac 1);
 by (dres_inst_tac [("y","X n")] isLubD2 1);
-by (ALLGOALS(blast_tac (claset() addDs [real_le_anti_sym])));
+by (ALLGOALS(blast_tac (claset() addDs [order_antisym])));
 qed "lemma_converg1";
 
 (*------------------------------------------------------------------- 
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/Star.thy
--- a/src/HOL/Hyperreal/Star.thy	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/Star.thy	Fri Mar 19 10:51:03 2004 +0100
@@ -180,7 +180,6 @@
 *)
 
 lemma hrabs_is_starext_rabs: "is_starext abs abs"
-
 apply (simp add: is_starext_def, safe)
 apply (rule_tac z = x in eq_Abs_hypreal)
 apply (rule_tac z = y in eq_Abs_hypreal, auto)
@@ -190,9 +189,10 @@
 apply (arith | ultra)+
 done
 
-lemma Rep_hypreal_FreeUltrafilterNat: "[| X \<in> Rep_hypreal z; Y \<in> Rep_hypreal z |]
+lemma Rep_hypreal_FreeUltrafilterNat:
+     "[| X \<in> Rep_hypreal z; Y \<in> Rep_hypreal z |]
       ==> {n. X n = Y n} : FreeUltrafilterNat"
-apply (rule_tac z = z in eq_Abs_hypreal)
+apply (cases z)
 apply (auto, ultra)
 done
 
@@ -212,22 +212,25 @@
                  UN_equiv_class [OF equiv_hyprel starfun_congruent])
 done
 
+lemma starfun_if_eq:
+     "w \<noteq> hypreal_of_real x
+       ==> ( *f* (\<lambda>z. if z = x then a else g z)) w = ( *f* g) w" 
+apply (cases w) 
+apply (simp add: hypreal_of_real_def starfun, ultra)
+done
+
 (*-------------------------------------------
   multiplication: ( *f) x ( *g) = *(f x g)
  ------------------------------------------*)
 lemma starfun_mult: "( *f* f) xa * ( *f* g) xa = ( *f* (%x. f x * g x)) xa"
-apply (rule_tac z = xa in eq_Abs_hypreal)
-apply (auto simp add: starfun hypreal_mult)
-done
+by (cases xa, simp add: starfun hypreal_mult)
 declare starfun_mult [symmetric, simp]
 
 (*---------------------------------------
   addition: ( *f) + ( *g) = *(f + g)
  ---------------------------------------*)
 lemma starfun_add: "( *f* f) xa + ( *f* g) xa = ( *f* (%x. f x + g x)) xa"
-apply (rule_tac z = xa in eq_Abs_hypreal)
-apply (auto simp add: starfun hypreal_add)
-done
+by (cases xa, simp add: starfun hypreal_add)
 declare starfun_add [symmetric, simp]
 
 (*--------------------------------------------
@@ -235,7 +238,7 @@
  -------------------------------------------*)
 
 lemma starfun_minus: "- ( *f* f) x = ( *f* (%x. - f x)) x"
-apply (rule_tac z = x in eq_Abs_hypreal)
+apply (cases x)
 apply (auto simp add: starfun hypreal_minus)
 done
 declare starfun_minus [symmetric, simp]
@@ -271,7 +274,7 @@
   NS extension of constant function
  --------------------------------------*)
 lemma starfun_const_fun: "( *f* (%x. k)) xa = hypreal_of_real  k"
-apply (rule_tac z = xa in eq_Abs_hypreal)
+apply (cases xa)
 apply (auto simp add: starfun hypreal_of_real_def)
 done
 
@@ -282,12 +285,12 @@
  ----------------------------------------------------*)
 
 lemma starfun_Idfun_approx: "x @= hypreal_of_real a ==> ( *f* (%x. x)) x @= hypreal_of_real  a"
-apply (rule_tac z = x in eq_Abs_hypreal)
+apply (cases x)
 apply (auto simp add: starfun)
 done
 
 lemma starfun_Id: "( *f* (%x. x)) x = x"
-apply (rule_tac z = x in eq_Abs_hypreal)
+apply (cases x)
 apply (auto simp add: starfun)
 done
 declare starfun_Id [simp]
@@ -297,7 +300,6 @@
  ----------------------------------------------------------------------*)
 
 lemma is_starext_starfun: "is_starext ( *f* f) f"
-
 apply (simp add: is_starext_def, auto)
 apply (rule_tac z = x in eq_Abs_hypreal)
 apply (rule_tac z = y in eq_Abs_hypreal)
@@ -309,7 +311,6 @@
  ----------------------------------------------------------------------*)
 
 lemma is_starfun_starext: "is_starext F f ==> F = *f* f"
-
 apply (simp add: is_starext_def)
 apply (rule ext)
 apply (rule_tac z = x in eq_Abs_hypreal)
@@ -336,12 +337,12 @@
 
 (* useful for NS definition of derivatives *)
 lemma starfun_lambda_cancel: "( *f* (%h. f (x + h))) xa  = ( *f* f) (hypreal_of_real  x + xa)"
-apply (rule_tac z = xa in eq_Abs_hypreal)
+apply (cases xa)
 apply (auto simp add: starfun hypreal_of_real_def hypreal_add)
 done
 
 lemma starfun_lambda_cancel2: "( *f* (%h. f(g(x + h)))) xa = ( *f* (f o g)) (hypreal_of_real x + xa)"
-apply (rule_tac z = xa in eq_Abs_hypreal)
+apply (cases xa)
 apply (auto simp add: starfun hypreal_of_real_def hypreal_add)
 done
 
@@ -370,13 +371,13 @@
 by (rule hrabs_is_starext_rabs [THEN is_starext_starfun_iff [THEN iffD1], symmetric])
 
 lemma starfun_inverse_inverse: "( *f* inverse) x = inverse(x)"
-apply (rule_tac z = x in eq_Abs_hypreal)
+apply (cases x)
 apply (auto simp add: starfun hypreal_inverse hypreal_zero_def)
 done
 declare starfun_inverse_inverse [simp]
 
 lemma starfun_inverse: "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x"
-apply (rule_tac z = x in eq_Abs_hypreal)
+apply (cases x)
 apply (auto simp add: starfun hypreal_inverse)
 done
 declare starfun_inverse [symmetric, simp]
@@ -386,7 +387,7 @@
 declare starfun_divide [symmetric, simp]
 
 lemma starfun_inverse2: "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x"
-apply (rule_tac z = x in eq_Abs_hypreal)
+apply (cases x)
 apply (auto intro: FreeUltrafilterNat_subset dest!: FreeUltrafilterNat_Compl_mem simp add: starfun hypreal_inverse hypreal_zero_def)
 done
 
@@ -397,7 +398,7 @@
 lemma starfun_mem_starset:
       "( *f* f) x : *s* A ==> x : *s* {x. f x  \<in> A}"
 apply (simp add: starset_def)
-apply (rule_tac z = x in eq_Abs_hypreal)
+apply (cases x)
 apply (auto simp add: starfun)
 apply (rename_tac "X")
 apply (drule_tac x = "%n. f (X n) " in bspec)
@@ -446,7 +447,7 @@
       (\<exists>X \<in> Rep_hypreal(x).
         \<forall>m. {n. abs(X n) < inverse(real(Suc m))}
                 \<in>  FreeUltrafilterNat)"
-apply (rule eq_Abs_hypreal [of x])
+apply (cases x)
 apply (auto intro!: bexI lemma_hyprel_refl 
             simp add: Infinitesimal_hypreal_of_nat_iff hypreal_of_real_def
      hypreal_inverse hypreal_hrabs hypreal_less hypreal_of_nat_eq)
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/Hyperreal/Transcendental.ML
--- a/src/HOL/Hyperreal/Transcendental.ML	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/Hyperreal/Transcendental.ML	Fri Mar 19 10:51:03 2004 +0100
@@ -5,6 +5,9 @@
     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" 
@@ -272,7 +275,7 @@
 (*-------------------------------------------------------------------------*)
 
 Goal "summable (%n. inverse (real (fact n)) * x ^ n)";
-by (cut_facts_tac [zero_less_one RS real_dense] 1);
+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;
@@ -727,7 +730,7 @@
 by (asm_full_simp_tac (simpset() addsimps [real_diff_def,
      right_distrib] @ add_ac @ mult_ac) 1);
 (* 46 *)
-by (dtac real_dense 1 THEN Step_tac 1);
+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))"),
@@ -2551,7 +2554,7 @@
 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 [real_le_anti_sym],simpset()));
+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;
diff -r 758e7acdea2f -r cc61fd03e589 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Fri Mar 19 10:50:06 2004 +0100
+++ b/src/HOL/IsaMakefile	Fri Mar 19 10:51:03 2004 +0100
@@ -148,7 +148,7 @@
   Hyperreal/HTranscendental.thy Hyperreal/HyperArith.thy\
   Hyperreal/HyperDef.thy Hyperreal/HyperNat.thy\
   Hyperreal/HyperPow.thy Hyperreal/Hyperreal.thy Hyperreal/IntFloor.thy\
-  Hyperreal/Lim.ML Hyperreal/Lim.thy Hyperreal/Log.thy\
+  Hyperreal/Lim.thy Hyperreal/Log.thy\
   Hyperreal/MacLaurin.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\