src/HOL/Hyperreal/Lim.thy
author paulson
Mon, 04 Oct 2004 15:28:03 +0200
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permissions -rw-r--r--
revised simprules for division
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(*  Title       : Lim.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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header{*Limits, Continuity and Differentiation*}
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theory Lim
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imports SEQ RealDef
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begin
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text{*Standard and Nonstandard Definitions*}
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constdefs
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  LIM :: "[real=>real,real,real] => bool"
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				("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60)
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  "f -- a --> L ==
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     \<forall>r. 0 < r -->
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	     (\<exists>s. 0 < s & (\<forall>x. (x \<noteq> a & (\<bar>x + -a\<bar> < s)
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			  --> \<bar>f x + -L\<bar> < r)))"
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  NSLIM :: "[real=>real,real,real] => bool"
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			      ("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60)
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  "f -- a --NS> L == (\<forall>x. (x \<noteq> hypreal_of_real a &
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		      x @= hypreal_of_real a -->
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		      ( *f* f) x @= hypreal_of_real L))"
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  isCont :: "[real=>real,real] => bool"
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  "isCont f a == (f -- a --> (f a))"
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  isNSCont :: "[real=>real,real] => bool"
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    --{*NS definition dispenses with limit notions*}
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  "isNSCont f a == (\<forall>y. y @= hypreal_of_real a -->
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			   ( *f* f) y @= hypreal_of_real (f a))"
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  deriv:: "[real=>real,real,real] => bool"
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    --{*Differentiation: D is derivative of function f at x*}
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			    ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
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  "DERIV f x :> D == ((%h. (f(x + h) + -f x)/h) -- 0 --> D)"
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  nsderiv :: "[real=>real,real,real] => bool"
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			    ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
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  "NSDERIV f x :> D == (\<forall>h \<in> Infinitesimal - {0}.
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			(( *f* f)(hypreal_of_real x + h) +
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			 - hypreal_of_real (f x))/h @= hypreal_of_real D)"
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  differentiable :: "[real=>real,real] => bool"   (infixl "differentiable" 60)
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  "f differentiable x == (\<exists>D. DERIV f x :> D)"
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  NSdifferentiable :: "[real=>real,real] => bool"   
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                       (infixl "NSdifferentiable" 60)
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  "f NSdifferentiable x == (\<exists>D. NSDERIV f x :> D)"
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  increment :: "[real=>real,real,hypreal] => hypreal"
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  "increment f x h == (@inc. f NSdifferentiable x &
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		       inc = ( *f* f)(hypreal_of_real x + h) + -hypreal_of_real (f x))"
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  isUCont :: "(real=>real) => bool"
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  "isUCont f ==  (\<forall>r. 0 < r -->
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		      (\<exists>s. 0 < s & (\<forall>x y. \<bar>x + -y\<bar> < s
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			    --> \<bar>f x + -f y\<bar> < r)))"
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  isNSUCont :: "(real=>real) => bool"
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  "isNSUCont f == (\<forall>x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
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consts
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  Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
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    --{*Used in the proof of the Bolzano theorem*}
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primrec
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  "Bolzano_bisect P a b 0 = (a,b)"
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  "Bolzano_bisect P a b (Suc n) =
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      (let (x,y) = Bolzano_bisect P a b n
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       in if P(x, (x+y)/2) then ((x+y)/2, y)
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                            else (x, (x+y)/2))"
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section{*Some Purely Standard Proofs*}
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lemma LIM_eq:
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     "f -- a --> L =
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     (\<forall>r. 0<r --> (\<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)))"
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by (simp add: LIM_def diff_minus)
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lemma LIM_D:
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     "[| f -- a --> L; 0<r |]
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      ==> \<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)"
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by (simp add: LIM_eq)
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lemma LIM_const [simp]: "(%x. k) -- x --> k"
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by (simp add: LIM_def)
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lemma LIM_add:
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  assumes f: "f -- a --> L" and g: "g -- a --> M"
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  shows "(%x. f x + g(x)) -- a --> (L + M)"
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proof (simp add: LIM_eq, clarify)
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  fix r :: real
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  assume r: "0<r"
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  from LIM_D [OF f half_gt_zero [OF r]]
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  obtain fs
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    where fs:    "0 < fs"
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      and fs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < fs --> \<bar>f x - L\<bar> < r/2"
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  by blast
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  from LIM_D [OF g half_gt_zero [OF r]]
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  obtain gs
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    where gs:    "0 < gs"
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      and gs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < gs --> \<bar>g x - M\<bar> < r/2"
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  by blast
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  show "\<exists>s. 0 < s \<and>
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            (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> \<bar>f x + g x - (L + M)\<bar> < r)"
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  proof (intro exI conjI strip)
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    show "0 < min fs gs"  by (simp add: fs gs)
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    fix x :: real
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    assume "x \<noteq> a \<and> \<bar>x-a\<bar> < min fs gs"
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    with fs_lt gs_lt
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    have "\<bar>f x - L\<bar> < r/2" and "\<bar>g x - M\<bar> < r/2" by (auto simp add: fs_lt)
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    hence "\<bar>f x - L\<bar> + \<bar>g x - M\<bar> < r" by arith
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    thus "\<bar>f x + g x - (L + M)\<bar> < r"
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      by (blast intro: abs_diff_triangle_ineq order_le_less_trans)
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  qed
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qed
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lemma LIM_minus: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
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apply (simp add: LIM_eq)
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apply (subgoal_tac "\<forall>x. \<bar>- f x + L\<bar> = \<bar>f x - L\<bar>")
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apply (simp_all add: abs_if)
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done
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lemma LIM_add_minus:
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    "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
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by (blast dest: LIM_add LIM_minus)
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lemma LIM_diff:
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    "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) - g(x)) -- x --> l-m"
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by (simp add: diff_minus LIM_add_minus) 
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lemma LIM_const_not_eq: "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
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proof (simp add: linorder_neq_iff LIM_eq, elim disjE)
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  assume k: "k < L"
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  show "\<exists>r. 0 < r \<and>
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        (\<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)"
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  proof (intro exI conjI strip)
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    show "0 < L-k" by (simp add: k compare_rls)
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    fix s :: real
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    assume s: "0<s"
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    { from s show "s/2 + a < a \<or> a < s/2 + a" by arith
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     next
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      from s show "\<bar>s / 2 + a - a\<bar> < s" by (simp add: abs_if) 
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     next
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   156
      from s show "~ \<bar>k-L\<bar> < L-k" by (simp add: abs_if) }
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   157
  qed
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   158
next
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   159
  assume k: "L < k"
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   160
  show "\<exists>r. 0 < r \<and>
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        (\<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)"
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   162
  proof (intro exI conjI strip)
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   163
    show "0 < k-L" by (simp add: k compare_rls)
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   164
    fix s :: real
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   165
    assume s: "0<s"
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   166
    { from s show "s/2 + a < a \<or> a < s/2 + a" by arith
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   167
     next
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   168
      from s show "\<bar>s / 2 + a - a\<bar> < s" by (simp add: abs_if) 
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   169
     next
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      from s show "~ \<bar>k-L\<bar> < k-L" by (simp add: abs_if) }
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   171
  qed
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   172
qed
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   173
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lemma LIM_const_eq: "(%x. k) -- x --> L ==> k = L"
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   175
apply (rule ccontr)
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apply (blast dest: LIM_const_not_eq) 
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   177
done
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   178
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lemma LIM_unique: "[| f -- a --> L; f -- a --> M |] ==> L = M"
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apply (drule LIM_diff, assumption) 
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apply (auto dest!: LIM_const_eq)
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   182
done
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   183
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   184
lemma LIM_mult_zero:
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  assumes f: "f -- a --> 0" and g: "g -- a --> 0"
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   186
  shows "(%x. f(x) * g(x)) -- a --> 0"
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proof (simp add: LIM_eq, clarify)
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  fix r :: real
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  assume r: "0<r"
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  from LIM_D [OF f zero_less_one]
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  obtain fs
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    where fs:    "0 < fs"
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      and fs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < fs --> \<bar>f x\<bar> < 1"
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   194
  by auto
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   195
  from LIM_D [OF g r]
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   196
  obtain gs
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   197
    where gs:    "0 < gs"
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   198
      and gs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < gs --> \<bar>g x\<bar> < r"
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   199
  by auto
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   200
  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)"
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   201
  proof (intro exI conjI strip)
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   202
    show "0 < min fs gs"  by (simp add: fs gs)
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   203
    fix x :: real
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   204
    assume "x \<noteq> a \<and> \<bar>x-a\<bar> < min fs gs"
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   205
    with fs_lt gs_lt
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   206
    have "\<bar>f x\<bar> < 1" and "\<bar>g x\<bar> < r" by (auto simp add: fs_lt)
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   207
    hence "\<bar>f x\<bar> * \<bar>g x\<bar> < 1*r" by (rule abs_mult_less) 
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   208
    thus "\<bar>f x\<bar> * \<bar>g x\<bar> < r" by simp
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   209
  qed
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   210
qed
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   211
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   212
lemma LIM_self: "(%x. x) -- a --> a"
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   213
by (auto simp add: LIM_def)
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   214
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text{*Limits are equal for functions equal except at limit point*}
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lemma LIM_equal:
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     "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
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   218
by (simp add: LIM_def)
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   219
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text{*Two uses in Hyperreal/Transcendental.ML*}
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lemma LIM_trans:
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     "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
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   223
apply (drule LIM_add, assumption)
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   224
apply (auto simp add: add_assoc)
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   225
done
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   226
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   227
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   228
subsection{*Relationships Between Standard and Nonstandard Concepts*}
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text{*Standard and NS definitions of Limit*} (*NEEDS STRUCTURING*)
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lemma LIM_NSLIM:
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      "f -- x --> L ==> f -- x --NS> L"
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   233
apply (simp add: LIM_def NSLIM_def approx_def)
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   234
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
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   235
apply (rule_tac z = xa in eq_Abs_hypreal)
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   236
apply (auto simp add: real_add_minus_iff starfun hypreal_minus hypreal_of_real_def hypreal_add)
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   237
apply (rule bexI, rule_tac [2] lemma_hyprel_refl, clarify) 
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   238
apply (drule_tac x = u in spec, clarify)
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   239
apply (drule_tac x = s in spec, clarify)
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   240
apply (subgoal_tac "\<forall>n::nat. (xa n) \<noteq> x & \<bar>(xa n) + - x\<bar> < s --> \<bar>f (xa n) + - L\<bar> < u")
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   241
prefer 2 apply blast
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   242
apply (drule FreeUltrafilterNat_all, ultra)
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   243
done
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   244
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   245
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   246
subsubsection{*Limit: The NS definition implies the standard definition.*}
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   247
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   248
lemma lemma_LIM: "\<forall>s. 0 < s --> (\<exists>xa.  xa \<noteq> x &
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   249
         \<bar>xa + - x\<bar> < s  & r \<le> \<bar>f xa + -L\<bar>)
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   250
      ==> \<forall>n::nat. \<exists>xa.  xa \<noteq> x &
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   251
              \<bar>xa + -x\<bar> < inverse(real(Suc n)) & r \<le> \<bar>f xa + -L\<bar>"
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   252
apply clarify
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   253
apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
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   254
done
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   255
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   256
lemma lemma_skolemize_LIM2:
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   257
     "\<forall>s. 0 < s --> (\<exists>xa.  xa \<noteq> x &
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   258
         \<bar>xa + - x\<bar> < s  & r \<le> \<bar>f xa + -L\<bar>)
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   259
      ==> \<exists>X. \<forall>n::nat. X n \<noteq> x &
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   260
                \<bar>X n + -x\<bar> < inverse(real(Suc n)) & r \<le> abs(f (X n) + -L)"
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   261
apply (drule lemma_LIM)
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   262
apply (drule choice, blast)
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   263
done
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   264
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   265
lemma lemma_simp: "\<forall>n. X n \<noteq> x &
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   266
          \<bar>X n + - x\<bar> < inverse (real(Suc n)) &
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   267
          r \<le> abs (f (X n) + - L) ==>
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   268
          \<forall>n. \<bar>X n + - x\<bar> < inverse (real(Suc n))"
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   269
by auto
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   270
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   271
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text{*NSLIM => LIM*}
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   273
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   274
lemma NSLIM_LIM: "f -- x --NS> L ==> f -- x --> L"
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   275
apply (simp add: LIM_def NSLIM_def approx_def)
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   276
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, clarify)
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   277
apply (rule ccontr, simp)
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   278
apply (simp add: linorder_not_less)
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   279
apply (drule lemma_skolemize_LIM2, safe)
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   280
apply (drule_tac x = "Abs_hypreal (hyprel``{X}) " in spec)
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   281
apply (auto simp add: starfun hypreal_minus hypreal_of_real_def hypreal_add)
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diff changeset
   282
apply (drule lemma_simp [THEN real_seq_to_hypreal_Infinitesimal])
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diff changeset
   283
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff hypreal_of_real_def hypreal_minus hypreal_add, blast)
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   284
apply (drule spec, drule mp, assumption)
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   285
apply (drule FreeUltrafilterNat_all, ultra)
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   286
done
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parents: 14387
diff changeset
   287
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   288
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   289
theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
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   290
by (blast intro: LIM_NSLIM NSLIM_LIM)
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diff changeset
   291
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   292
text{*Proving properties of limits using nonstandard definition.
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   293
      The properties hold for standard limits as well!*}
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   294
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   295
lemma NSLIM_mult:
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     "[| f -- x --NS> l; g -- x --NS> m |]
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      ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
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by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
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   299
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lemma LIM_mult2:
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     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) * g(x)) -- x --> (l * m)"
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   302
by (simp add: LIM_NSLIM_iff NSLIM_mult)
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   303
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lemma NSLIM_add:
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     "[| f -- x --NS> l; g -- x --NS> m |]
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      ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
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   307
by (auto simp add: NSLIM_def intro!: approx_add)
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   308
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lemma LIM_add2:
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     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + g(x)) -- x --> (l + m)"
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   311
by (simp add: LIM_NSLIM_iff NSLIM_add)
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   312
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   313
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lemma NSLIM_const [simp]: "(%x. k) -- x --NS> k"
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   315
by (simp add: NSLIM_def)
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   316
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lemma LIM_const2: "(%x. k) -- x --> k"
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   318
by (simp add: LIM_NSLIM_iff)
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   319
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   320
lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
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   321
by (simp add: NSLIM_def)
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   322
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   323
lemma LIM_minus2: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
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   324
by (simp add: LIM_NSLIM_iff NSLIM_minus)
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   325
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   326
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lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
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   328
by (blast dest: NSLIM_add NSLIM_minus)
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   329
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   330
lemma LIM_add_minus2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
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   331
by (simp add: LIM_NSLIM_iff NSLIM_add_minus)
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   332
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   333
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   334
lemma NSLIM_inverse:
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     "[| f -- a --NS> L;  L \<noteq> 0 |]
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      ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
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   337
apply (simp add: NSLIM_def, clarify)
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   338
apply (drule spec)
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   339
apply (auto simp add: hypreal_of_real_approx_inverse)
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   340
done
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   341
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   342
lemma LIM_inverse: "[| f -- a --> L; L \<noteq> 0 |] ==> (%x. inverse(f(x))) -- a --> (inverse L)"
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   343
by (simp add: LIM_NSLIM_iff NSLIM_inverse)
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   344
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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   345
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   346
lemma NSLIM_zero:
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  assumes f: "f -- a --NS> l" shows "(%x. f(x) + -l) -- a --NS> 0"
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   348
proof -
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   349
  have "(\<lambda>x. f x + - l) -- a --NS> l + -l"
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   350
    by (rule NSLIM_add_minus [OF f NSLIM_const])
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   351
  thus ?thesis by simp
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   352
qed
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   353
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   354
lemma LIM_zero2: "f -- a --> l ==> (%x. f(x) + -l) -- a --> 0"
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   355
by (simp add: LIM_NSLIM_iff NSLIM_zero)
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diff changeset
   356
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   357
lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
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   358
apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
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   359
apply (auto simp add: diff_minus add_assoc)
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   360
done
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diff changeset
   361
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   362
lemma LIM_zero_cancel: "(%x. f(x) - l) -- x --> 0 ==> f -- x --> l"
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   363
apply (drule_tac g = "%x. l" and M = l in LIM_add)
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   364
apply (auto simp add: diff_minus add_assoc)
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   365
done
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diff changeset
   366
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   367
lemma NSLIM_not_zero: "k \<noteq> 0 ==> ~ ((%x. k) -- x --NS> 0)"
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   368
apply (simp add: NSLIM_def)
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   369
apply (rule_tac x = "hypreal_of_real x + epsilon" in exI)
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   370
apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
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   371
            simp add: hypreal_epsilon_not_zero)
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   372
done
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   373
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   374
lemma NSLIM_const_not_eq: "k \<noteq> L ==> ~ ((%x. k) -- x --NS> L)"
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   375
apply (simp add: NSLIM_def)
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   376
apply (rule_tac x = "hypreal_of_real x + epsilon" in exI)
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   377
apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
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   378
            simp add: hypreal_epsilon_not_zero)
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   379
done
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   380
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   381
lemma NSLIM_const_eq: "(%x. k) -- x --NS> L ==> k = L"
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   382
apply (rule ccontr)
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   383
apply (blast dest: NSLIM_const_not_eq)
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   384
done
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   385
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   386
text{* can actually be proved more easily by unfolding the definition!*}
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   387
lemma NSLIM_unique: "[| f -- x --NS> L; f -- x --NS> M |] ==> L = M"
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   388
apply (drule NSLIM_minus)
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   389
apply (drule NSLIM_add, assumption)
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   390
apply (auto dest!: NSLIM_const_eq [symmetric])
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diff changeset
   391
done
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diff changeset
   392
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   393
lemma LIM_unique2: "[| f -- x --> L; f -- x --> M |] ==> L = M"
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   394
by (simp add: LIM_NSLIM_iff NSLIM_unique)
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diff changeset
   395
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   396
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   397
lemma NSLIM_mult_zero: "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
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   398
by (drule NSLIM_mult, auto)
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diff changeset
   399
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   400
(* we can use the corresponding thm LIM_mult2 *)
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diff changeset
   401
(* for standard definition of limit           *)
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diff changeset
   402
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   403
lemma LIM_mult_zero2: "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0"
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diff changeset
   404
by (drule LIM_mult2, auto)
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diff changeset
   405
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   406
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diff changeset
   407
lemma NSLIM_self: "(%x. x) -- a --NS> a"
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diff changeset
   408
by (simp add: NSLIM_def)
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diff changeset
   409
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   410
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   411
subsection{* Derivatives and Continuity: NS and Standard properties*}
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diff changeset
   412
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diff changeset
   413
subsubsection{*Continuity*}
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   414
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   415
lemma isNSContD: "[| isNSCont f a; y \<approx> hypreal_of_real a |] ==> ( *f* f) y \<approx> hypreal_of_real (f a)"
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diff changeset
   416
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   417
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   418
lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
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diff changeset
   419
by (simp add: isNSCont_def NSLIM_def)
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diff changeset
   420
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   421
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
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diff changeset
   422
apply (simp add: isNSCont_def NSLIM_def, auto)
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diff changeset
   423
apply (rule_tac Q = "y = hypreal_of_real a" in excluded_middle [THEN disjE], auto)
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diff changeset
   424
done
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diff changeset
   425
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   426
text{*NS continuity can be defined using NS Limit in
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   427
    similar fashion to standard def of continuity*}
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diff changeset
   428
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
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diff changeset
   429
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
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diff changeset
   430
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diff changeset
   431
text{*Hence, NS continuity can be given
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   432
  in terms of standard limit*}
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diff changeset
   433
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
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paulson
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diff changeset
   434
by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
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diff changeset
   435
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diff changeset
   436
text{*Moreover, it's trivial now that NS continuity
4d332d10fa3d revised simprules for division
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diff changeset
   437
  is equivalent to standard continuity*}
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diff changeset
   438
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
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diff changeset
   439
apply (simp add: isCont_def)
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parents: 14387
diff changeset
   440
apply (rule isNSCont_LIM_iff)
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diff changeset
   441
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   442
15228
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diff changeset
   443
text{*Standard continuity ==> NS continuity*}
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diff changeset
   444
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
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diff changeset
   445
by (erule isNSCont_isCont_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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parents: 14387
diff changeset
   446
15228
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diff changeset
   447
text{*NS continuity ==> Standard continuity*}
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diff changeset
   448
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
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diff changeset
   449
by (erule isNSCont_isCont_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   450
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   451
text{*Alternative definition of continuity*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   452
(* Prove equivalence between NS limits - *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   453
(* seems easier than using standard def  *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   454
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   455
apply (simp add: NSLIM_def, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   456
apply (drule_tac x = "hypreal_of_real a + x" in spec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   457
apply (drule_tac [2] x = "-hypreal_of_real a + x" in spec, safe, simp)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   458
apply (rule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   459
apply (rule_tac [4] approx_minus_iff2 [THEN iffD1])
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   460
 prefer 3 apply (simp add: add_commute)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   461
apply (rule_tac [2] z = x in eq_Abs_hypreal)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   462
apply (rule_tac [4] z = x in eq_Abs_hypreal)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   463
apply (auto simp add: starfun hypreal_of_real_def hypreal_minus hypreal_add add_assoc approx_refl hypreal_zero_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   464
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   465
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   466
lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   467
by (rule NSLIM_h_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   468
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   469
lemma LIM_isCont_iff: "(f -- a --> f a) = ((%h. f(a + h)) -- 0 --> f(a))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   470
by (simp add: LIM_NSLIM_iff NSLIM_isCont_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   471
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   472
lemma isCont_iff: "(isCont f x) = ((%h. f(x + h)) -- 0 --> f(x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   473
by (simp add: isCont_def LIM_isCont_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   474
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   475
text{*Immediate application of nonstandard criterion for continuity can offer
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   476
   very simple proofs of some standard property of continuous functions*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   477
text{*sum continuous*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   478
lemma isCont_add: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   479
by (auto intro: approx_add simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   480
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   481
text{*mult continuous*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   482
lemma isCont_mult: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   483
by (auto intro!: starfun_mult_HFinite_approx
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   484
            simp del: starfun_mult [symmetric]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   485
            simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   486
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   487
text{*composition of continuous functions
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   488
     Note very short straightforard proof!*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   489
lemma isCont_o: "[| isCont f a; isCont g (f a) |] ==> isCont (g o f) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   490
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_def starfun_o [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   491
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   492
lemma isCont_o2: "[| isCont f a; isCont g (f a) |] ==> isCont (%x. g (f x)) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   493
by (auto dest: isCont_o simp add: o_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   494
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   495
lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   496
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   497
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   498
lemma isCont_minus: "isCont f a ==> isCont (%x. - f x) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   499
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   500
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   501
lemma isCont_inverse:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   502
      "[| isCont f x; f x \<noteq> 0 |] ==> isCont (%x. inverse (f x)) x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   503
apply (simp add: isCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   504
apply (blast intro: LIM_inverse)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   505
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   506
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   507
lemma isNSCont_inverse: "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   508
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   509
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   510
lemma isCont_diff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   511
      "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   512
apply (simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   513
apply (auto intro: isCont_add isCont_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   514
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   515
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   516
lemma isCont_const [simp]: "isCont (%x. k) a"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   517
by (simp add: isCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   518
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   519
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   520
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   521
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   522
lemma isNSCont_abs [simp]: "isNSCont abs a"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   523
apply (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   524
apply (auto intro: approx_hrabs simp add: hypreal_of_real_hrabs [symmetric] starfun_rabs_hrabs)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   525
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   526
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   527
lemma isCont_abs [simp]: "isCont abs a"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   528
by (auto simp add: isNSCont_isCont_iff [symmetric])
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   529
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   530
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   531
(****************************************************************
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   532
(%* Leave as commented until I add topology theory or remove? *%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   533
(%*------------------------------------------------------------
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   534
  Elementary topology proof for a characterisation of
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   535
  continuity now: a function f is continuous if and only
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   536
  if the inverse image, {x. f(x) \<in> A}, of any open set A
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   537
  is always an open set
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   538
 ------------------------------------------------------------*%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   539
Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   540
               ==> isNSopen {x. f x \<in> A}"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   541
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   542
by (dtac (mem_monad_approx RS approx_sym);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   543
by (dres_inst_tac [("x","a")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   544
by (dtac isNSContD 1 THEN assume_tac 1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   545
by (dtac bspec 1 THEN assume_tac 1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   546
by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   547
by (blast_tac (claset() addIs [starfun_mem_starset]);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   548
qed "isNSCont_isNSopen";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   549
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   550
Goalw [isNSCont_def]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   551
          "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   552
\              ==> isNSCont f x";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   553
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   554
     (approx_minus_iff RS iffD2)],simpset() addsimps
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   555
      [Infinitesimal_def,SReal_iff]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   556
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   557
by (etac (isNSopen_open_interval RSN (2,impE));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   558
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   559
by (dres_inst_tac [("x","x")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   560
by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   561
    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   562
qed "isNSopen_isNSCont";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   563
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   564
Goal "(\<forall>x. isNSCont f x) = \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   565
\     (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   566
by (blast_tac (claset() addIs [isNSCont_isNSopen,
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   567
    isNSopen_isNSCont]);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   568
qed "isNSCont_isNSopen_iff";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   569
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   570
(%*------- Standard version of same theorem --------*%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   571
Goal "(\<forall>x. isCont f x) = \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   572
\         (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   573
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   574
              simpset() addsimps [isNSopen_isopen_iff RS sym,
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   575
              isNSCont_isCont_iff RS sym]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   576
qed "isCont_isopen_iff";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   577
*******************************************************************)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   578
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   579
text{*Uniform continuity*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   580
lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   581
by (simp add: isNSUCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   582
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   583
lemma isUCont_isCont: "isUCont f ==> isCont f x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   584
by (simp add: isUCont_def isCont_def LIM_def, meson)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   585
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   586
lemma isUCont_isNSUCont: "isUCont f ==> isNSUCont f"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   587
apply (simp add: isNSUCont_def isUCont_def approx_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   588
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   589
apply (rule_tac z = x in eq_Abs_hypreal)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   590
apply (rule_tac z = y in eq_Abs_hypreal)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   591
apply (auto simp add: starfun hypreal_minus hypreal_add)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   592
apply (rule bexI, rule_tac [2] lemma_hyprel_refl, safe)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   593
apply (drule_tac x = u in spec, clarify)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   594
apply (drule_tac x = s in spec, clarify)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   595
apply (subgoal_tac "\<forall>n::nat. abs ((xa n) + - (xb n)) < s --> abs (f (xa n) + - f (xb n)) < u")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   596
prefer 2 apply blast
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   597
apply (erule_tac V = "\<forall>x y. \<bar>x + - y\<bar> < s --> \<bar>f x + - f y\<bar> < u" in thin_rl)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   598
apply (drule FreeUltrafilterNat_all, ultra)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   599
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   600
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   601
lemma lemma_LIMu: "\<forall>s. 0 < s --> (\<exists>z y. \<bar>z + - y\<bar> < s & r \<le> \<bar>f z + -f y\<bar>)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   602
      ==> \<forall>n::nat. \<exists>z y.
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   603
               \<bar>z + -y\<bar> < inverse(real(Suc n)) &
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   604
               r \<le> \<bar>f z + -f y\<bar>"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   605
apply clarify
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   606
apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   607
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   608
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   609
lemma lemma_skolemize_LIM2u: "\<forall>s. 0 < s --> (\<exists>z y. \<bar>z + - y\<bar> < s  & r \<le> \<bar>f z + -f y\<bar>)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   610
      ==> \<exists>X Y. \<forall>n::nat.
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   611
               abs(X n + -(Y n)) < inverse(real(Suc n)) &
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   612
               r \<le> abs(f (X n) + -f (Y n))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   613
apply (drule lemma_LIMu)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   614
apply (drule choice, safe)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   615
apply (drule choice, blast)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   616
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   617
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   618
lemma lemma_simpu: "\<forall>n. \<bar>X n + -Y n\<bar> < inverse (real(Suc n)) &
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   619
          r \<le> abs (f (X n) + - f(Y n)) ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   620
          \<forall>n. \<bar>X n + - Y n\<bar> < inverse (real(Suc n))"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   621
by auto
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   622
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   623
lemma isNSUCont_isUCont:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   624
     "isNSUCont f ==> isUCont f"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   625
apply (simp add: isNSUCont_def isUCont_def approx_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   626
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff, safe)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   627
apply (rule ccontr, simp)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   628
apply (simp add: linorder_not_less)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   629
apply (drule lemma_skolemize_LIM2u, safe)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   630
apply (drule_tac x = "Abs_hypreal (hyprel``{X}) " in spec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   631
apply (drule_tac x = "Abs_hypreal (hyprel``{Y}) " in spec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   632
apply (simp add: starfun hypreal_minus hypreal_add, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   633
apply (drule lemma_simpu [THEN real_seq_to_hypreal_Infinitesimal2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   634
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff hypreal_minus hypreal_add, blast)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   635
apply (drule_tac x = r in spec, clarify)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   636
apply (drule FreeUltrafilterNat_all, ultra)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   637
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   638
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   639
text{*Derivatives*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   640
lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   641
by (simp add: deriv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   642
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   643
lemma DERIV_NS_iff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   644
      "(DERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --NS> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   645
by (simp add: deriv_def LIM_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   646
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   647
lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) + - f(x))/h) -- 0 --> D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   648
by (simp add: deriv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   649
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   650
lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) + - f(x))/h) -- 0 --NS> D"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   651
by (simp add: deriv_def LIM_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   652
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   653
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   654
subsubsection{*Uniqueness*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   655
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   656
lemma DERIV_unique:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   657
      "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   658
apply (simp add: deriv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   659
apply (blast intro: LIM_unique)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   660
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   661
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   662
lemma NSDeriv_unique:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   663
     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   664
apply (simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   665
apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   666
apply (auto dest!: bspec [where x=epsilon]
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   667
            intro!: inj_hypreal_of_real [THEN injD]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   668
            dest: approx_trans3)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   669
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   670
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   671
subsubsection{*Differentiable*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   672
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   673
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   674
by (simp add: differentiable_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   675
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   676
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   677
by (force simp add: differentiable_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   678
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   679
lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   680
by (simp add: NSdifferentiable_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   681
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   682
lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   683
by (force simp add: NSdifferentiable_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   684
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   685
subsubsection{*Alternative definition for differentiability*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   686
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   687
lemma LIM_I:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   688
     "(!!r. 0<r ==> (\<exists>s. 0 < s & (\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>f x - L\<bar> < r)))
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   689
      ==> f -- a --> L"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   690
by (simp add: LIM_eq)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   691
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   692
lemma DERIV_LIM_iff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   693
     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   694
      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   695
proof (intro iffI LIM_I)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   696
  fix r::real
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   697
  assume r: "0<r"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   698
  assume "(\<lambda>h. (f (a + h) - f a) / h) -- 0 --> D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   699
  from LIM_D [OF this r]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   700
  obtain s
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   701
    where s:    "0 < s"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   702
      and s_lt: "\<forall>x. x \<noteq> 0 & \<bar>x\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   703
  by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   704
  show "\<exists>s. 0 < s \<and>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   705
            (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> \<bar>(f x - f a) / (x-a) - D\<bar> < r)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   706
  proof (intro exI conjI strip)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   707
    show "0 < s"  by (rule s)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   708
  next
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   709
    fix x::real
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   710
    assume "x \<noteq> a \<and> \<bar>x-a\<bar> < s"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   711
    with s_lt [THEN spec [where x="x-a"]]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   712
    show "\<bar>(f x - f a) / (x-a) - D\<bar> < r" by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   713
  qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   714
next
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   715
  fix r::real
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   716
  assume r: "0<r"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   717
  assume "(\<lambda>x. (f x - f a) / (x-a)) -- a --> D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   718
  from LIM_D [OF this r]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   719
  obtain s
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   720
    where s:    "0 < s"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   721
      and s_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>(f x - f a)/(x-a) - D\<bar> < r"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   722
  by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   723
  show "\<exists>s. 0 < s \<and>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   724
            (\<forall>x. x \<noteq> 0 & \<bar>x - 0\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   725
  proof (intro exI conjI strip)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   726
    show "0 < s"  by (rule s)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   727
  next
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   728
    fix x::real
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   729
    assume "x \<noteq> 0 \<and> \<bar>x - 0\<bar> < s"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   730
    with s_lt [THEN spec [where x="x+a"]]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   731
    show "\<bar>(f (a + x) - f a) / x - D\<bar> < r" by (auto simp add: add_ac)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   732
  qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   733
qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   734
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   735
lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   736
by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   737
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   738
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   739
subsection{*Equivalence of NS and standard definitions of differentiation*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   740
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   741
subsubsection{*First NSDERIV in terms of NSLIM*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   742
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   743
text{*first equivalence *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   744
lemma NSDERIV_NSLIM_iff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   745
      "(NSDERIV f x :> D) = ((%h. (f(x + h) + - f(x))/h) -- 0 --NS> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   746
apply (simp add: nsderiv_def NSLIM_def, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   747
apply (drule_tac x = xa in bspec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   748
apply (rule_tac [3] ccontr)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   749
apply (drule_tac [3] x = h in spec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   750
apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   751
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   752
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   753
text{*second equivalence *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   754
lemma NSDERIV_NSLIM_iff2:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   755
     "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   756
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff  diff_minus [symmetric]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   757
              LIM_NSLIM_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   758
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   759
(* while we're at it! *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   760
lemma NSDERIV_iff2:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   761
     "(NSDERIV f x :> D) =
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   762
      (\<forall>w.
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   763
        w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   764
        ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   765
by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   766
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   767
(*FIXME DELETE*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   768
lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   769
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   770
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   771
lemma NSDERIVD5:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   772
  "(NSDERIV f x :> D) ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   773
   (\<forall>u. u \<approx> hypreal_of_real x -->
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   774
     ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   775
apply (auto simp add: NSDERIV_iff2)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   776
apply (case_tac "u = hypreal_of_real x", auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   777
apply (drule_tac x = u in spec, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   778
apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   779
apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   780
apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   781
apply (auto simp add: diff_minus
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   782
	       approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   783
		     Infinitesimal_subset_HFinite [THEN subsetD])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   784
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   785
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   786
lemma NSDERIVD4:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   787
     "(NSDERIV f x :> D) ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   788
      (\<forall>h \<in> Infinitesimal.
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   789
               (( *f* f)(hypreal_of_real x + h) -
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   790
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   791
apply (auto simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   792
apply (case_tac "h = (0::hypreal) ")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   793
apply (auto simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   794
apply (drule_tac x = h in bspec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   795
apply (drule_tac [2] c = h in approx_mult1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   796
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   797
            simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   798
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   799
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   800
lemma NSDERIVD3:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   801
     "(NSDERIV f x :> D) ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   802
      (\<forall>h \<in> Infinitesimal - {0}.
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   803
               (( *f* f)(hypreal_of_real x + h) -
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   804
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   805
apply (auto simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   806
apply (rule ccontr, drule_tac x = h in bspec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   807
apply (drule_tac [2] c = h in approx_mult1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   808
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   809
            simp add: mult_assoc diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   810
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   811
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   812
text{*Now equivalence between NSDERIV and DERIV*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   813
lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   814
by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   815
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   816
text{*Differentiability implies continuity
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   817
         nice and simple "algebraic" proof*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   818
lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   819
apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   820
apply (drule approx_minus_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   821
apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   822
apply (drule_tac x = "-hypreal_of_real x + xa" in bspec)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   823
 prefer 2 apply (simp add: add_assoc [symmetric])
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   824
apply (auto simp add: mem_infmal_iff [symmetric] hypreal_add_commute)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   825
apply (drule_tac c = "xa + -hypreal_of_real x" in approx_mult1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   826
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   827
            simp add: mult_assoc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   828
apply (drule_tac x3=D in
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   829
           HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   830
             THEN mem_infmal_iff [THEN iffD1]])
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   831
apply (auto simp add: mult_commute
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   832
            intro: approx_trans approx_minus_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   833
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   834
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   835
text{*Now Sandard proof*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   836
lemma DERIV_isCont: "DERIV f x :> D ==> isCont f x"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   837
by (simp add: NSDERIV_DERIV_iff [symmetric] isNSCont_isCont_iff [symmetric]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   838
              NSDERIV_isNSCont)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   839
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   840
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   841
text{*Differentiation rules for combinations of functions
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   842
      follow from clear, straightforard, algebraic
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   843
      manipulations*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   844
text{*Constant function*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   845
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   846
(* use simple constant nslimit theorem *)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   847
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   848
by (simp add: NSDERIV_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   849
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   850
lemma DERIV_const [simp]: "(DERIV (%x. k) x :> 0)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   851
by (simp add: NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   852
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   853
text{*Sum of functions- proved easily*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   854
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   855
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   856
lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   857
      ==> NSDERIV (%x. f x + g x) x :> Da + Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   858
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   859
apply (auto simp add: add_divide_distrib dest!: spec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   860
apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   861
apply (auto simp add: add_ac)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   862
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   863
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   864
(* Standard theorem *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   865
lemma DERIV_add: "[| DERIV f x :> Da; DERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   866
      ==> DERIV (%x. f x + g x) x :> Da + Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   867
apply (simp add: NSDERIV_add NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   868
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   869
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   870
text{*Product of functions - Proof is trivial but tedious
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   871
  and long due to rearrangement of terms*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   872
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   873
lemma lemma_nsderiv1: "((a::hypreal)*b) + -(c*d) = (b*(a + -c)) + (c*(b + -d))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   874
by (simp add: right_distrib)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   875
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   876
lemma lemma_nsderiv2: "[| (x + y) / z = hypreal_of_real D + yb; z \<noteq> 0;
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   877
         z \<in> Infinitesimal; yb \<in> Infinitesimal |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   878
      ==> x + y \<approx> 0"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   879
apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   880
apply (erule_tac V = " (x + y) / z = hypreal_of_real D + yb" in thin_rl)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   881
apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   882
            simp add: hypreal_mult_assoc mem_infmal_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   883
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   884
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   885
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   886
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   887
lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   888
      ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   889
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   890
apply (auto dest!: spec
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   891
	    simp add: starfun_lambda_cancel lemma_nsderiv1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   892
apply (simp (no_asm) add: add_divide_distrib)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   893
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   894
apply (auto simp del: times_divide_eq_right simp add: times_divide_eq_right [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   895
apply (drule_tac D = Db in lemma_nsderiv2)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   896
apply (drule_tac [4]
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   897
     approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   898
apply (auto intro!: approx_add_mono1
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   899
            simp add: left_distrib right_distrib mult_commute add_assoc)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   900
apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   901
         in add_commute [THEN subst])
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   902
apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   903
                    Infinitesimal_add Infinitesimal_mult
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   904
                    Infinitesimal_hypreal_of_real_mult
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   905
                    Infinitesimal_hypreal_of_real_mult2
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   906
          simp add: add_assoc [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   907
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   908
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   909
lemma DERIV_mult:
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   910
     "[| DERIV f x :> Da; DERIV g x :> Db |]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   911
      ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   912
by (simp add: NSDERIV_mult NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   913
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   914
text{*Multiplying by a constant*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   915
lemma NSDERIV_cmult: "NSDERIV f x :> D
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   916
      ==> NSDERIV (%x. c * f x) x :> c*D"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   917
apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   918
                  minus_mult_right right_distrib [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   919
apply (erule NSLIM_const [THEN NSLIM_mult])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   920
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   921
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   922
(* let's do the standard proof though theorem *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   923
(* LIM_mult2 follows from a NS proof          *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   924
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   925
lemma DERIV_cmult:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   926
      "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   927
apply (simp only: deriv_def times_divide_eq_right [symmetric]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   928
                  NSDERIV_NSLIM_iff minus_mult_right right_distrib [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   929
apply (erule LIM_const [THEN LIM_mult2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   930
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   931
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   932
text{*Negation of function*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   933
lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   934
proof (simp add: NSDERIV_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   935
  assume "(\<lambda>h. (f (x + h) + - f x) / h) -- 0 --NS> D"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   936
  hence deriv: "(\<lambda>h. - ((f(x+h) + - f x) / h)) -- 0 --NS> - D"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   937
    by (rule NSLIM_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   938
  have "\<forall>h. - ((f (x + h) + - f x) / h) = (- f (x + h) + f x) / h"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   939
    by (simp add: minus_divide_left)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   940
  with deriv
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   941
  show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   942
qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   943
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   944
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   945
lemma DERIV_minus: "DERIV f x :> D ==> DERIV (%x. -(f x)) x :> -D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   946
by (simp add: NSDERIV_minus NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   947
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   948
text{*Subtraction*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   949
lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   950
by (blast dest: NSDERIV_add NSDERIV_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   951
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   952
lemma DERIV_add_minus: "[| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x + -g x) x :> Da + -Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   953
by (blast dest: DERIV_add DERIV_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   954
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   955
lemma NSDERIV_diff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   956
     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   957
      ==> NSDERIV (%x. f x - g x) x :> Da-Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   958
apply (simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   959
apply (blast intro: NSDERIV_add_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   960
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   961
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   962
lemma DERIV_diff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   963
     "[| DERIV f x :> Da; DERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   964
       ==> DERIV (%x. f x - g x) x :> Da-Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   965
apply (simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   966
apply (blast intro: DERIV_add_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   967
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   968
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   969
text{*(NS) Increment*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   970
lemma incrementI:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   971
      "f NSdifferentiable x ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   972
      increment f x h = ( *f* f) (hypreal_of_real(x) + h) +
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   973
      -hypreal_of_real (f x)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   974
by (simp add: increment_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   975
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   976
lemma incrementI2: "NSDERIV f x :> D ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   977
     increment f x h = ( *f* f) (hypreal_of_real(x) + h) +
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   978
     -hypreal_of_real (f x)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   979
apply (erule NSdifferentiableI [THEN incrementI])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   980
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   981
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   982
(* The Increment theorem -- Keisler p. 65 *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   983
lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   984
      ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   985
apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   986
apply (drule bspec, auto)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   987
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   988
apply (frule_tac b1 = "hypreal_of_real (D) + y"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   989
        in hypreal_mult_right_cancel [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   990
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)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   991
apply assumption
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   992
apply (simp add: times_divide_eq_right [symmetric] del: times_divide_eq_right)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   993
apply (auto simp add: left_distrib)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   994
done
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   995
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   996
lemma increment_thm2:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   997
     "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   998
      ==> \<exists>e \<in> Infinitesimal. increment f x h =
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   999
              hypreal_of_real(D)*h + e*h"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1000
by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1001
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1002
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1003
lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1004
      ==> increment f x h \<approx> 0"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1005
apply (drule increment_thm2,
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1006
       auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1007
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1008
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1009
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1010
text{*  Similarly to the above, the chain rule admits an entirely
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1011
   straightforward derivation. Compare this with Harrison's
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1012
   HOL proof of the chain rule, which proved to be trickier and
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1013
   required an alternative characterisation of differentiability-
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1014
   the so-called Carathedory derivative. Our main problem is
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1015
   manipulation of terms.*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1016
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1017
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1018
(* lemmas *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1019
lemma NSDERIV_zero:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1020
      "[| NSDERIV g x :> D;
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1021
               ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1022
               xa \<in> Infinitesimal;
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1023
               xa \<noteq> 0
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1024
            |] ==> D = 0"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1025
apply (simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1026
apply (drule bspec, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1027
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1028
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1029
(* can be proved differently using NSLIM_isCont_iff *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1030
lemma NSDERIV_approx:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1031
     "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1032
      ==> ( *f* f) (hypreal_of_real(x) + h) + -hypreal_of_real(f x) \<approx> 0"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1033
apply (simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1034
apply (simp add: mem_infmal_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1035
apply (rule Infinitesimal_ratio)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1036
apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1037
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1038
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1039
(*---------------------------------------------------------------
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1040
   from one version of differentiability
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1041
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1042
                f(x) - f(a)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1043
              --------------- \<approx> Db
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1044
                  x - a
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1045
 ---------------------------------------------------------------*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1046
lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1047
         ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1048
         ( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1049
      |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1050
                   + - hypreal_of_real (f (g x)))
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1051
              / (( *f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real (g x))
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1052
             \<approx> hypreal_of_real(Da)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1053
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1054
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1055
(*--------------------------------------------------------------
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1056
   from other version of differentiability
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1057
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1058
                f(x + h) - f(x)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1059
               ----------------- \<approx> Db
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1060
                       h
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1061
 --------------------------------------------------------------*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1062
lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1063
      ==> (( *f* g) (hypreal_of_real(x) + xa) + - hypreal_of_real(g x)) / xa
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1064
          \<approx> hypreal_of_real(Db)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1065
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1066
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1067
lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1068
by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1069
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1070
text{*This proof uses both definitions of differentiability.*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1071
lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1072
      ==> NSDERIV (f o g) x :> Da * Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1073
apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1074
                mem_infmal_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1075
apply clarify
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1076
apply (frule_tac f = g in NSDERIV_approx)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1077
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1078
apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1079
apply (drule_tac g = g in NSDERIV_zero)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1080
apply (auto simp add: divide_inverse)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1081
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])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1082
apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1083
apply (rule approx_mult_hypreal_of_real)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1084
apply (simp_all add: divide_inverse [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1085
apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1086
apply (blast intro: NSDERIVD2)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1087
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1088
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1089
(* standard version *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1090
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1091
by (simp add: NSDERIV_DERIV_iff [symmetric] NSDERIV_chain)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1092
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1093
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1094
by (auto dest: DERIV_chain simp add: o_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1095
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1096
text{*Differentiation of natural number powers*}
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1097
lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1098
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1099
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1100
(*derivative of the identity function*)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1101
lemma DERIV_Id [simp]: "DERIV (%x. x) x :> 1"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1102
by (simp add: NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1103
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1104
lemmas isCont_Id = DERIV_Id [THEN DERIV_isCont, standard]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1105
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1106
(*derivative of linear multiplication*)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1107
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1108
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1109
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1110
lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1111
by (simp add: NSDERIV_DERIV_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1112
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1113
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1114
apply (induct_tac "n")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1115
apply (drule_tac [2] DERIV_Id [THEN DERIV_mult])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1116
apply (auto simp add: real_of_nat_Suc left_distrib)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1117
apply (case_tac "0 < n")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1118
apply (drule_tac x = x in realpow_minus_mult)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1119
apply (auto simp add: real_mult_assoc real_add_commute)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1120
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1121
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1122
(* NS version *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1123
lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1124
by (simp add: NSDERIV_DERIV_iff DERIV_pow)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1125
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1126
text{*Power of -1*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1127
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1128
(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1129
lemma NSDERIV_inverse:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1130
     "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1131
apply (simp add: nsderiv_def)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1132
apply (rule ballI, simp, clarify)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1133
apply (frule Infinitesimal_add_not_zero)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1134
prefer 2 apply (simp add: add_commute)
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1135
apply (auto simp add: starfun_inverse_inverse realpow_two
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1136
        simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1137
apply (simp add: inverse_add inverse_mult_distrib [symmetric]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1138
              inverse_minus_eq [symmetric] add_ac mult_ac
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1139
            del: inverse_mult_distrib inverse_minus_eq
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1140
                 minus_mult_left [symmetric] minus_mult_right [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1141
apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1142
            del: minus_mult_left [symmetric] minus_mult_right [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1143
apply (rule_tac y = " inverse (- hypreal_of_real x * hypreal_of_real x) " in approx_trans)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1144
apply (rule inverse_add_Infinitesimal_approx2)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1145
apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1146
            simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1147
apply (rule Infinitesimal_HFinite_mult2, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1148
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1149
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1150
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1151
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1152
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1153
lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1154
by (simp add: NSDERIV_inverse NSDERIV_DERIV_iff [symmetric] del: realpow_Suc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1155
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1156
text{*Derivative of inverse*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1157
lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1158
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1159
apply (simp only: mult_commute [of d] minus_mult_left power_inverse)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1160
apply (fold o_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1161
apply (blast intro!: DERIV_chain DERIV_inverse)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1162
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1163
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1164
lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1165
      ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1166
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1167
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1168
text{*Derivative of quotient*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1169
lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1170
       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) + -(e*f(x))) / (g(x) ^ Suc (Suc 0))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1171
apply (drule_tac f = g in DERIV_inverse_fun)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1172
apply (drule_tac [2] DERIV_mult)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1173
apply (assumption+)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1174
apply (simp add: divide_inverse right_distrib power_inverse minus_mult_left
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1175
                 mult_ac
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1176
     del: realpow_Suc minus_mult_right [symmetric] minus_mult_left [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1177
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1178
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1179
lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1180
       ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1181
                            + -(e*f(x))) / (g(x) ^ Suc (Suc 0))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1182
by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1183
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1184
(* ------------------------------------------------------------------------ *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1185
(* Caratheodory formulation of derivative at a point: standard proof        *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1186
(* ------------------------------------------------------------------------ *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1187
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1188
lemma CARAT_DERIV:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1189
     "(DERIV f x :> l) =
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1190
      (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1191
      (is "?lhs = ?rhs")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1192
proof
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1193
  assume der: "DERIV f x :> l"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1194
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1195
  proof (intro exI conjI)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1196
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1197
    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1198
    show "isCont ?g x" using der
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1199
      by (simp add: isCont_iff DERIV_iff diff_minus
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1200
               cong: LIM_equal [rule_format])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1201
    show "?g x = l" by simp
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1202
  qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1203
next
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1204
  assume "?rhs"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1205
  then obtain g where
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1206
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1207
  thus "(DERIV f x :> l)"
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1208
     by (auto simp add: isCont_iff DERIV_iff diff_minus
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1209
               cong: LIM_equal [rule_format])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1210
qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1211
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1212
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1213
lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1214
      \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1215
by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1216
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1217
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1218
by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1219
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1220
lemma CARAT_DERIVD:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1221
  assumes all: "\<forall>z. f z - f x = g z * (z-x)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1222
      and nsc: "isNSCont g x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1223
  shows "NSDERIV f x :> g x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1224
proof -
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1225
  from nsc
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1226
  have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1227
         ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1228
         hypreal_of_real (g x)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1229
    by (simp add: diff_minus isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1230
  thus ?thesis using all
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1231
    by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1232
qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1233
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1234
(*--------------------------------------------------------------------------*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1235
(* Lemmas about nested intervals and proof by bisection (cf.Harrison)       *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1236
(* All considerably tidied by lcp                                           *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1237
(*--------------------------------------------------------------------------*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1238
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1239
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)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1240
apply (induct_tac "no")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1241
apply (auto intro: order_trans)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1242
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1243
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1244
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1245
         \<forall>n. g(Suc n) \<le> g(n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1246
         \<forall>n. f(n) \<le> g(n) |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1247
      ==> Bseq f"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1248
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1249
apply (induct_tac "n")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1250
apply (auto intro: order_trans)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1251
apply (rule_tac y = "g (Suc na) " in order_trans)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1252
apply (induct_tac [2] "na")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1253
apply (auto intro: order_trans)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1254
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1255
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1256
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1257
         \<forall>n. g(Suc n) \<le> g(n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1258
         \<forall>n. f(n) \<le> g(n) |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1259
      ==> Bseq g"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1260
apply (subst Bseq_minus_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1261
apply (rule_tac g = "%x. - (f x) " in f_inc_g_dec_Beq_f)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1262
apply auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1263
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1264
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1265
lemma f_inc_imp_le_lim: "[| \<forall>n. f n \<le> f (Suc n);  convergent f |] ==> f n \<le> lim f"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1266
apply (rule linorder_not_less [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1267
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1268
apply (drule real_less_sum_gt_zero)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1269
apply (drule_tac x = "f n + - lim f" in spec, safe)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1270
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1271
apply (subgoal_tac "lim f \<le> f (no + n) ")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1272
apply (induct_tac [2] "no")
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14477
diff changeset
  1273
apply (auto intro: order_trans simp add: diff_minus abs_if)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1274
apply (drule_tac no=no and m=n in lemma_f_mono_add)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1275
apply (auto simp add: add_commute)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1276
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1277
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1278
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1279
apply (rule LIMSEQ_minus [THEN limI])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1280
apply (simp add: convergent_LIMSEQ_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1281
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1282
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1283
lemma g_dec_imp_lim_le: "[| \<forall>n. g(Suc n) \<le> g(n);  convergent g |] ==> lim g \<le> g n"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1284
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1285
apply (cut_tac [2] f = "%x. - (g x) " in f_inc_imp_le_lim)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1286
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1287
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1288
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1289
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1290
         \<forall>n. g(Suc n) \<le> g(n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1291
         \<forall>n. f(n) \<le> g(n) |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1292
      ==> \<exists>l m. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1293
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1294
apply (subgoal_tac "monoseq f & monoseq g")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1295
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1296
apply (subgoal_tac "Bseq f & Bseq g")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1297
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1298
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1299
apply (rule_tac x = "lim f" in exI)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1300
apply (rule_tac x = "lim g" in exI)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1301
apply (auto intro: LIMSEQ_le)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1302
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1303
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1304
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1305
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1306
         \<forall>n. g(Suc n) \<le> g(n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1307
         \<forall>n. f(n) \<le> g(n);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1308
         (%n. f(n) - g(n)) ----> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: