src/HOL/Hyperreal/Lim.thy

simplification tweaks for better arithmetic reasoning

(*  Title       : Lim.thy
    ID          : $Id$
    Author      : Jacques D. Fleuriot
    Copyright   : 1998  University of Cambridge
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)

header{*Limits, Continuity and Differentiation*}

theory Lim
imports SEQ RealDef
begin

text{*Standard and Nonstandard Definitions*}

constdefs
  LIM :: "[real=>real,real,real] => bool"
				("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60)
  "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)))"

  NSLIM :: "[real=>real,real,real] => bool"
			      ("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60)
  "f -- a --NS> L == (\<forall>x. (x \<noteq> hypreal_of_real a &
		      x @= hypreal_of_real a -->
		      ( *f* f) x @= hypreal_of_real L))"

  isCont :: "[real=>real,real] => bool"
  "isCont f a == (f -- a --> (f a))"

  isNSCont :: "[real=>real,real] => bool"
    --{*NS definition dispenses with limit notions*}
  "isNSCont f a == (\<forall>y. y @= hypreal_of_real a -->
			   ( *f* f) y @= hypreal_of_real (f a))"

  deriv:: "[real=>real,real,real] => bool"
    --{*Differentiation: D is derivative of function f at x*}
			    ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
  "DERIV f x :> D == ((%h. (f(x + h) + -f x)/h) -- 0 --> D)"

  nsderiv :: "[real=>real,real,real] => bool"
			    ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
  "NSDERIV f x :> D == (\<forall>h \<in> Infinitesimal - {0}.
			(( *f* f)(hypreal_of_real x + h) +
			 - hypreal_of_real (f x))/h @= hypreal_of_real D)"

  differentiable :: "[real=>real,real] => bool"   (infixl "differentiable" 60)
  "f differentiable x == (\<exists>D. DERIV 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 &
		       inc = ( *f* f)(hypreal_of_real x + h) + -hypreal_of_real (f x))"

  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 == (\<forall>x y. x @= y --> ( *f* f) x @= ( *f* f) y)"


consts
  Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
    --{*Used in the proof of the Bolzano theorem*}


primrec
  "Bolzano_bisect P a b 0 = (a,b)"
  "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))"



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 [simp]: "(%x. k) -- x --> k"
by (simp add: LIM_def)

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 compare_rls)
    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 compare_rls)
    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 & \<bar>(xa n) + - x\<bar> < s --> \<bar>f (xa n) + - L\<bar> < u")
prefer 2 apply blast
apply (drule FreeUltrafilterNat_all, ultra)
done


subsubsection{*Limit: The NS definition implies the 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


text{*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


theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
by (blast intro: LIM_NSLIM NSLIM_LIM)

text{*Proving properties of limits using nonstandard definition.
      The properties hold for standard limits as well!*}

lemma NSLIM_mult:
     "[| f -- x --NS> l; g -- x --NS> m |]
      ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)

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)

lemma NSLIM_add:
     "[| f -- x --NS> l; g -- x --NS> m |]
      ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
by (auto simp add: NSLIM_def intro!: approx_add)

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 [simp]: "(%x. k) -- x --NS> k"
by (simp add: NSLIM_def)

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

text{* can actually be proved more easily by unfolding the definition!*}
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)


subsection{* Derivatives and Continuity: NS and Standard properties*}

subsubsection{*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

text{*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)

text{*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)

text{*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

text{*Standard continuity ==> NS continuity*}
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
by (erule isNSCont_isCont_iff [THEN iffD2])

text{*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)

text{*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)

text{*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 [simp]: "isCont (%x. k) a"
by (simp add: isCont_def)

lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
by (simp add: isNSCont_def)

lemma isNSCont_abs [simp]: "isNSCont abs a"
apply (simp add: isNSCont_def)
apply (auto intro: approx_hrabs simp add: hypreal_of_real_hrabs [symmetric] starfun_rabs_hrabs)
done

lemma isCont_abs [simp]: "isCont abs a"
by (auto simp add: isNSCont_isCont_iff [symmetric])


(****************************************************************
(%* 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))"
by auto

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 (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*}

subsubsection{*First NSDERIV in terms of NSLIM*}

text{*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

text{*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: times_divide_eq_left 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: times_divide_eq_left 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 times_divide_eq_left 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)

text{*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] 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: times_divide_eq_right 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)


text{*Differentiation rules for combinations of functions
      follow from clear, straightforard, algebraic
      manipulations*}
text{*Constant function*}

(* use simple constant nslimit theorem *)
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
by (simp add: NSDERIV_NSLIM_iff)

lemma DERIV_const [simp]: "(DERIV (%x. k) x :> 0)"
by (simp add: NSDERIV_DERIV_iff [symmetric])

text{*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

text{*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: times_divide_eq_left 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 add: times_divide_eq_right [symmetric]
            simp del: times_divide_eq)
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

text{*(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_left times_divide_eq_right [symmetric])
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

text{*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 [simp]: "NSDERIV (%x. x) x :> 1"
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)

(*derivative of the identity function*)
lemma DERIV_Id [simp]: "DERIV (%x. x) x :> 1"
by (simp add: NSDERIV_DERIV_iff [symmetric])

lemmas isCont_Id = DERIV_Id [THEN DERIV_isCont, standard]

(*derivative of linear multiplication*)
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)

lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
by (simp add: NSDERIV_DERIV_iff)

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: mult_assoc 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)

text{*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 add: times_divide_eq)
    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

text{*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 abs_if)
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