src/HOL/Hyperreal/Deriv.thy
author huffman
Tue, 12 Dec 2006 04:37:25 +0100
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(*  Title       : Deriv.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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    GMVT by Benjamin Porter, 2005
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*)
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header{* Differentiation *}
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theory Deriv
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imports Lim
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begin
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text{*Standard and Nonstandard Definitions*}
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definition
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  deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
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    --{*Differentiation: D is derivative of function f at x*}
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          ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
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  "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"
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definition
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  nsderiv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
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          ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
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  "NSDERIV f x :> D = (\<forall>h \<in> Infinitesimal - {0}.
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      (( *f* f)(star_of x + h)
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       - star_of (f x))/h @= star_of D)"
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definition
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  differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
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    (infixl "differentiable" 60) where
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  "f differentiable x = (\<exists>D. DERIV f x :> D)"
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definition
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  NSdifferentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
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    (infixl "NSdifferentiable" 60) where
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  "f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)"
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definition
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  increment :: "[real=>real,real,hypreal] => hypreal" where
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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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consts
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  Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
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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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subsection {* Derivatives *}
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subsubsection {* Purely standard proofs *}
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lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
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by (simp add: deriv_def)
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lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
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by (simp add: deriv_def)
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lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
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by (simp add: deriv_def)
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lemma DERIV_Id [simp]: "DERIV (\<lambda>x. x) x :> 1"
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by (simp add: deriv_def divide_self cong: LIM_cong)
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lemma add_diff_add:
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  fixes a b c d :: "'a::ab_group_add"
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  shows "(a + c) - (b + d) = (a - b) + (c - d)"
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by simp
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lemma DERIV_add:
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  "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
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by (simp only: deriv_def add_diff_add add_divide_distrib LIM_add)
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lemma DERIV_minus:
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  "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
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by (simp only: deriv_def minus_diff_minus divide_minus_left LIM_minus)
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lemma DERIV_diff:
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  "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
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by (simp only: diff_def DERIV_add DERIV_minus)
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lemma DERIV_add_minus:
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  "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
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by (simp only: DERIV_add DERIV_minus)
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lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
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proof (unfold isCont_iff)
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  assume "DERIV f x :> D"
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  hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"
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    by (rule DERIV_D)
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  hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
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    by (intro LIM_mult LIM_self)
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  hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
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    by simp
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  hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
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    by (simp cong: LIM_cong add: divide_self)
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  thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
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    by (simp add: LIM_def)
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qed
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lemma DERIV_mult_lemma:
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  fixes a b c d :: "'a::real_field"
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  shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"
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by (simp add: diff_minus add_divide_distrib [symmetric] ring_distrib)
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lemma DERIV_mult':
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  assumes f: "DERIV f x :> D"
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  assumes g: "DERIV g x :> E"
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  shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
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proof (unfold deriv_def)
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  from f have "isCont f x"
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    by (rule DERIV_isCont)
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  hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
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    by (simp only: isCont_iff)
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  hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +
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              ((f(x+h) - f x) / h) * g x)
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          -- 0 --> f x * E + D * g x"
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    by (intro LIM_add LIM_mult2 LIM_const DERIV_D f g)
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  thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)
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         -- 0 --> f x * E + D * g x"
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    by (simp only: DERIV_mult_lemma)
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qed
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lemma DERIV_mult:
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     "[| DERIV f x :> Da; DERIV g x :> Db |]
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      ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
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by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
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lemma DERIV_unique:
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      "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
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apply (simp add: deriv_def)
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apply (blast intro: LIM_unique)
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done
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huffman
parents:
diff changeset
   141
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   142
text{*Differentiation of finite sum*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   143
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   144
lemma DERIV_sumr [rule_format (no_asm)]:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   145
     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   146
      --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   147
apply (induct "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   148
apply (auto intro: DERIV_add)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   149
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   150
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   151
text{*Alternative definition for differentiability*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   152
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   153
lemma DERIV_LIM_iff:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   154
     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   155
      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   156
apply (rule iffI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   157
apply (drule_tac k="- a" in LIM_offset)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   158
apply (simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   159
apply (drule_tac k="a" in LIM_offset)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   160
apply (simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   161
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   162
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   163
lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   164
by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   165
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   166
lemma inverse_diff_inverse:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   167
  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   168
   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   169
by (simp add: right_diff_distrib left_diff_distrib mult_assoc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   170
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   171
lemma DERIV_inverse_lemma:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   172
  "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   173
   \<Longrightarrow> (inverse a - inverse b) / h
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   174
     = - (inverse a * ((a - b) / h) * inverse b)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   175
by (simp add: inverse_diff_inverse)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   176
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   177
lemma DERIV_inverse':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   178
  assumes der: "DERIV f x :> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   179
  assumes neq: "f x \<noteq> 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   180
  shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   181
    (is "DERIV _ _ :> ?E")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   182
proof (unfold DERIV_iff2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   183
  from der have lim_f: "f -- x --> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   184
    by (rule DERIV_isCont [unfolded isCont_def])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   185
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   186
  from neq have "0 < norm (f x)" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   187
  with LIM_D [OF lim_f] obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   188
    where s: "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   189
    and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   190
                  \<Longrightarrow> norm (f z - f x) < norm (f x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   191
    by fast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   192
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   193
  show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   194
  proof (rule LIM_equal2 [OF s])
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   195
    fix z
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   196
    assume "z \<noteq> x" "norm (z - x) < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   197
    hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   198
    hence "f z \<noteq> 0" by auto
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   199
    thus "(inverse (f z) - inverse (f x)) / (z - x) =
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   200
          - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   201
      using neq by (rule DERIV_inverse_lemma)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   202
  next
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   203
    from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   204
      by (unfold DERIV_iff2)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   205
    thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   206
          -- x --> ?E"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   207
      by (intro LIM_mult2 LIM_inverse LIM_minus LIM_const lim_f neq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   208
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   209
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   210
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   211
lemma DERIV_divide:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   212
  "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   213
   \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   214
apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   215
          D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   216
apply (erule subst)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   217
apply (unfold divide_inverse)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   218
apply (erule DERIV_mult')
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   219
apply (erule (1) DERIV_inverse')
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   220
apply (simp add: left_diff_distrib nonzero_inverse_mult_distrib)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   221
apply (simp add: mult_ac)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   222
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   223
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   224
lemma DERIV_power_Suc:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   225
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   226
  assumes f: "DERIV f x :> D"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   227
  shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (of_nat n + 1) * (D * f x ^ n)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   228
proof (induct n)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   229
case 0
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   230
  show ?case by (simp add: power_Suc f)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   231
case (Suc k)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   232
  from DERIV_mult' [OF f Suc] show ?case
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   233
    apply (simp only: of_nat_Suc left_distrib mult_1_left)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   234
    apply (simp only: power_Suc right_distrib mult_ac)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   235
    done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   236
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   237
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   238
lemma DERIV_power:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   239
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   240
  assumes f: "DERIV f x :> D"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   241
  shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   242
by (cases "n", simp, simp add: DERIV_power_Suc f)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   243
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   244
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   245
(* ------------------------------------------------------------------------ *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   246
(* Caratheodory formulation of derivative at a point: standard proof        *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   247
(* ------------------------------------------------------------------------ *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   248
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   249
lemma nonzero_mult_divide_cancel_right:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   250
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   251
proof -
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   252
  assume b: "b \<noteq> 0"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   253
  have "a * b / b = a * (b / b)" by simp
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   254
  also have "\<dots> = a" by (simp add: divide_self b)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   255
  finally show "a * b / b = a" .
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   256
qed
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   257
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   258
lemma CARAT_DERIV:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   259
     "(DERIV f x :> l) =
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   260
      (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   261
      (is "?lhs = ?rhs")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   262
proof
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   263
  assume der: "DERIV f x :> l"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   264
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   265
  proof (intro exI conjI)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   266
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   267
    show "\<forall>z. f z - f x = ?g z * (z-x)"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   268
      by (simp add: nonzero_mult_divide_cancel_right)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   269
    show "isCont ?g x" using der
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   270
      by (simp add: isCont_iff DERIV_iff diff_minus
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   271
               cong: LIM_equal [rule_format])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   272
    show "?g x = l" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   273
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   274
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   275
  assume "?rhs"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   276
  then obtain g where
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   277
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   278
  thus "(DERIV f x :> l)"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   279
     by (auto simp add: isCont_iff DERIV_iff nonzero_mult_divide_cancel_right
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
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   280
              cong: LIM_cong)
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qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
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0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
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   283
lemma DERIV_chain':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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   284
  assumes f: "DERIV f x :> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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   285
  assumes g: "DERIV g (f x) :> E"
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parents: 21404
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   286
  shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
21164
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parents:
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   287
proof (unfold DERIV_iff2)
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parents: 21404
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   288
  obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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   289
    and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   290
    using CARAT_DERIV [THEN iffD1, OF g] by fast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
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   291
  from f have "f -- x --> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
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   292
    by (rule DERIV_isCont [unfolded isCont_def])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   293
  with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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parents: 21199
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   294
    by (rule isCont_LIM_compose)
21784
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parents: 21404
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   295
  hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
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   296
          -- x --> d (f x) * D"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
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   297
    by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
diff changeset
   298
  thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   299
    by (simp add: d dfx real_scaleR_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
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   300
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   301
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   302
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   303
subsubsection {* Nonstandard proofs *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
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   304
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
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   305
lemma DERIV_NS_iff:
21784
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parents: 21404
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   306
      "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   307
by (simp add: deriv_def LIM_NSLIM_iff)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
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   308
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
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   309
lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --NS> D"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   310
by (simp add: deriv_def LIM_NSLIM_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   311
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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   312
lemma hnorm_of_hypreal:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
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   313
  "\<And>r. hnorm (( *f* of_real) r::'a::real_normed_div_algebra star) = \<bar>r\<bar>"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
diff changeset
   314
by transfer (rule norm_of_real)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   315
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
diff changeset
   316
lemma Infinitesimal_of_hypreal:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
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   317
  "x \<in> Infinitesimal \<Longrightarrow>
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
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parents: 21404
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   318
   (( *f* of_real) x::'a::real_normed_div_algebra star) \<in> Infinitesimal"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   319
apply (rule InfinitesimalI2)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   320
apply (drule (1) InfinitesimalD2)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   321
apply (simp add: hnorm_of_hypreal)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   322
done
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   323
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   324
lemma of_hypreal_eq_0_iff:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   325
  "\<And>x. (( *f* of_real) x = (0::'a::real_algebra_1 star)) = (x = 0)"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   326
by transfer (rule of_real_eq_0_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   327
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   328
lemma NSDeriv_unique:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   329
     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   330
apply (subgoal_tac "( *f* of_real) epsilon \<in> Infinitesimal - {0::'a star}")
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   331
apply (simp only: nsderiv_def)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   332
apply (drule (1) bspec)+
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   333
apply (drule (1) approx_trans3)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   334
apply simp
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   335
apply (simp add: Infinitesimal_of_hypreal Infinitesimal_epsilon)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   336
apply (simp add: of_hypreal_eq_0_iff hypreal_epsilon_not_zero)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   337
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   338
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   339
text {*First NSDERIV in terms of NSLIM*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   340
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   341
text{*first equivalence *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   342
lemma NSDERIV_NSLIM_iff:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   343
      "(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   344
apply (simp add: nsderiv_def NSLIM_def, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   345
apply (drule_tac x = xa in bspec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   346
apply (rule_tac [3] ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   347
apply (drule_tac [3] x = h in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   348
apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   349
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   350
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   351
text{*second equivalence *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   352
lemma NSDERIV_NSLIM_iff2:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   353
     "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   354
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff  diff_minus [symmetric]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   355
              LIM_NSLIM_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   356
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   357
(* while we're at it! *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   358
lemma NSDERIV_iff2:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   359
     "(NSDERIV f x :> D) =
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   360
      (\<forall>w.
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   361
        w \<noteq> star_of x & w \<approx> star_of x -->
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   362
        ( *f* (%z. (f z - f x) / (z-x))) w \<approx> star_of D)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   363
by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   364
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   365
(*FIXME DELETE*)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   366
lemma hypreal_not_eq_minus_iff:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   367
  "(x \<noteq> a) = (x - a \<noteq> (0::'a::ab_group_add))"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   368
by auto
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   369
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   370
lemma NSDERIVD5:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   371
  "(NSDERIV f x :> D) ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   372
   (\<forall>u. u \<approx> hypreal_of_real x -->
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   373
     ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   374
apply (auto simp add: NSDERIV_iff2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   375
apply (case_tac "u = hypreal_of_real x", auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   376
apply (drule_tac x = u in spec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   377
apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   378
apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   379
apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   380
apply (auto simp add:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   381
         approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   382
         Infinitesimal_subset_HFinite [THEN subsetD])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   383
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   384
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   385
lemma NSDERIVD4:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   386
     "(NSDERIV f x :> D) ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   387
      (\<forall>h \<in> Infinitesimal.
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   388
               (( *f* f)(hypreal_of_real x + h) -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   389
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   390
apply (auto simp add: nsderiv_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   391
apply (case_tac "h = (0::hypreal) ")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   392
apply (auto simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   393
apply (drule_tac x = h in bspec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   394
apply (drule_tac [2] c = h in approx_mult1)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   395
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   396
            simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   397
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   398
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   399
lemma NSDERIVD3:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   400
     "(NSDERIV f x :> D) ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   401
      (\<forall>h \<in> Infinitesimal - {0}.
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   402
               (( *f* f)(hypreal_of_real x + h) -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   403
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   404
apply (auto simp add: nsderiv_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   405
apply (rule ccontr, drule_tac x = h in bspec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   406
apply (drule_tac [2] c = h in approx_mult1)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   407
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   408
            simp add: mult_assoc diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   409
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   410
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   411
text{*Differentiability implies continuity
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   412
         nice and simple "algebraic" proof*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   413
lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   414
apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   415
apply (drule approx_minus_iff [THEN iffD1])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   416
apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   417
apply (drule_tac x = "xa - star_of x" in bspec)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   418
 prefer 2 apply (simp add: add_assoc [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   419
apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   420
apply (drule_tac c = "xa - star_of x" in approx_mult1)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   421
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   422
            simp add: mult_assoc nonzero_mult_divide_cancel_right)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   423
apply (drule_tac x3=D in
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   424
           HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult,
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   425
             THEN mem_infmal_iff [THEN iffD1]])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   426
apply (auto simp add: mult_commute
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   427
            intro: approx_trans approx_minus_iff [THEN iffD2])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   428
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   429
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   430
text{*Differentiation rules for combinations of functions
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   431
      follow from clear, straightforard, algebraic
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   432
      manipulations*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   433
text{*Constant function*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   434
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   435
(* use simple constant nslimit theorem *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   436
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   437
by (simp add: NSDERIV_NSLIM_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   438
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   439
text{*Sum of functions- proved easily*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   440
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   441
lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   442
      ==> NSDERIV (%x. f x + g x) x :> Da + Db"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   443
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   444
apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   445
apply (drule_tac b = "star_of Da" and d = "star_of Db" in approx_add)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   446
apply (auto simp add: diff_def add_ac)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   447
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   448
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   449
text{*Product of functions - Proof is trivial but tedious
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   450
  and long due to rearrangement of terms*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   451
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   452
lemma lemma_nsderiv1:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   453
  fixes a b c d :: "'a::comm_ring star"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   454
  shows "(a*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   455
by (simp add: right_diff_distrib mult_ac)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   456
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   457
lemma lemma_nsderiv2:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   458
  fixes x y z :: "'a::real_normed_field star"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   459
  shows "[| (x - y) / z = star_of D + yb; z \<noteq> 0;
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   460
         z \<in> Infinitesimal; yb \<in> Infinitesimal |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   461
      ==> x - y \<approx> 0"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   462
apply (simp add: nonzero_divide_eq_eq)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   463
apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   464
            simp add: mult_assoc mem_infmal_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   465
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   466
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   467
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   468
lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   469
      ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   470
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   471
apply (auto dest!: spec
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   472
      simp add: starfun_lambda_cancel lemma_nsderiv1)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   473
apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   474
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   475
apply (auto simp add: times_divide_eq_right [symmetric]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   476
            simp del: times_divide_eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   477
apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   478
apply (drule_tac
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   479
     approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   480
apply (auto intro!: approx_add_mono1
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   481
            simp add: left_distrib right_distrib mult_commute add_assoc)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   482
apply (rule_tac b1 = "star_of Db * star_of (f x)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   483
         in add_commute [THEN subst])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   484
apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   485
                    Infinitesimal_add Infinitesimal_mult
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   486
                    Infinitesimal_hypreal_of_real_mult
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   487
                    Infinitesimal_hypreal_of_real_mult2
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   488
          simp add: add_assoc [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   489
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   490
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   491
text{*Multiplying by a constant*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   492
lemma NSDERIV_cmult: "NSDERIV f x :> D
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   493
      ==> NSDERIV (%x. c * f x) x :> c*D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   494
apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   495
                  minus_mult_right right_diff_distrib [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   496
apply (erule NSLIM_const [THEN NSLIM_mult])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   497
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   498
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   499
text{*Negation of function*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   500
lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   501
proof (simp add: NSDERIV_NSLIM_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   502
  assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   503
  hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   504
    by (rule NSLIM_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   505
  have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   506
    by (simp add: minus_divide_left diff_def)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   507
  with deriv
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   508
  show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   509
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   510
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   511
text{*Subtraction*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   512
lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   513
by (blast dest: NSDERIV_add NSDERIV_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   514
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   515
lemma NSDERIV_diff:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   516
     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   517
      ==> NSDERIV (%x. f x - g x) x :> Da-Db"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   518
apply (simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   519
apply (blast intro: NSDERIV_add_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   520
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   521
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   522
text{*  Similarly to the above, the chain rule admits an entirely
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   523
   straightforward derivation. Compare this with Harrison's
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   524
   HOL proof of the chain rule, which proved to be trickier and
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   525
   required an alternative characterisation of differentiability-
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   526
   the so-called Carathedory derivative. Our main problem is
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   527
   manipulation of terms.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   528
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   529
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   530
(* lemmas *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   531
lemma NSDERIV_zero:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   532
      "[| NSDERIV g x :> D;
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   533
               ( *f* g) (star_of x + xa) = star_of (g x);
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   534
               xa \<in> Infinitesimal;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   535
               xa \<noteq> 0
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   536
            |] ==> D = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   537
apply (simp add: nsderiv_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   538
apply (drule bspec, auto)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   539
apply (rule star_of_approx_iff [THEN iffD1], simp)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   540
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   541
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   542
(* can be proved differently using NSLIM_isCont_iff *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   543
lemma NSDERIV_approx:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   544
     "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   545
      ==> ( *f* f) (star_of x + h) - star_of (f x) \<approx> 0"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   546
apply (simp add: nsderiv_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   547
apply (simp add: mem_infmal_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   548
apply (rule Infinitesimal_ratio)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   549
apply (rule_tac [3] approx_star_of_HFinite, auto)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   550
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   551
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   552
(*---------------------------------------------------------------
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   553
   from one version of differentiability
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   554
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   555
                f(x) - f(a)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   556
              --------------- \<approx> Db
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   557
                  x - a
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   558
 ---------------------------------------------------------------*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   559
lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   560
         ( *f* g) (star_of(x) + xa) \<noteq> star_of (g x);
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   561
         ( *f* g) (star_of(x) + xa) \<approx> star_of (g x)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   562
      |] ==> (( *f* f) (( *f* g) (star_of(x) + xa))
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   563
                   - star_of (f (g x)))
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   564
              / (( *f* g) (star_of(x) + xa) - star_of (g x))
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   565
             \<approx> star_of(Da)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   566
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   567
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   568
(*--------------------------------------------------------------
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   569
   from other version of differentiability
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   570
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   571
                f(x + h) - f(x)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   572
               ----------------- \<approx> Db
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   573
                       h
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   574
 --------------------------------------------------------------*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   575
lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   576
      ==> (( *f* g) (star_of(x) + xa) - star_of(g x)) / xa
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   577
          \<approx> star_of(Db)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   578
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   579
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   580
lemma lemma_chain: "(z::'a::real_normed_field star) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   581
proof -
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   582
  assume z: "z \<noteq> 0"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   583
  have "x * y = x * (inverse z * z) * y" by (simp add: z)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   584
  thus ?thesis by (simp add: mult_assoc)
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   585
qed
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   586
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   587
text{*This proof uses both definitions of differentiability.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   588
lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   589
      ==> NSDERIV (f o g) x :> Da * Db"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   590
apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   591
                mem_infmal_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   592
apply clarify
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   593
apply (frule_tac f = g in NSDERIV_approx)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   594
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   595
apply (case_tac "( *f* g) (star_of (x) + xa) = star_of (g x) ")
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   596
apply (drule_tac g = g in NSDERIV_zero)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   597
apply (auto simp add: divide_inverse)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   598
apply (rule_tac z1 = "( *f* g) (star_of (x) + xa) - star_of (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst])
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   599
apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   600
apply (rule approx_mult_star_of)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   601
apply (simp_all add: divide_inverse [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   602
apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   603
apply (blast intro: NSDERIVD2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   604
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   605
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   606
text{*Differentiation of natural number powers*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   607
lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   608
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   609
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   610
lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   611
by (cut_tac c = c and x = x in NSDERIV_Id [THEN NSDERIV_cmult], simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   612
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   613
(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   614
lemma NSDERIV_inverse:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   615
  fixes x :: "'a::{real_normed_field,recpower}"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   616
  shows "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   617
apply (simp add: nsderiv_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   618
apply (rule ballI, simp, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   619
apply (frule (1) Infinitesimal_add_not_zero)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   620
apply (simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   621
(*apply (auto simp add: starfun_inverse_inverse realpow_two
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   622
        simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   623
apply (simp add: inverse_add nonzero_inverse_mult_distrib [symmetric] power_Suc
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   624
              nonzero_inverse_minus_eq [symmetric] add_ac mult_ac diff_def
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   625
            del: inverse_mult_distrib inverse_minus_eq
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   626
                 minus_mult_left [symmetric] minus_mult_right [symmetric])
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   627
apply (subst mult_commute, simp add: nonzero_mult_divide_cancel_right)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   628
apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   629
            del: minus_mult_left [symmetric] minus_mult_right [symmetric])
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   630
apply (rule_tac y = "inverse (- (star_of x * star_of x))" in approx_trans)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   631
apply (rule inverse_add_Infinitesimal_approx2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   632
apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   633
            simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   634
apply (rule Infinitesimal_HFinite_mult2, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   635
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   636
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   637
subsubsection {* Equivalence of NS and Standard definitions *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   638
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   639
lemma divideR_eq_divide: "x /# y = x / y"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   640
by (simp add: real_scaleR_def divide_inverse mult_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   641
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   642
text{*Now equivalence between NSDERIV and DERIV*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   643
lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   644
by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   645
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   646
(* let's do the standard proof though theorem *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   647
(* LIM_mult2 follows from a NS proof          *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   648
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   649
lemma DERIV_cmult:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   650
      "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   651
by (drule DERIV_mult' [OF DERIV_const], simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   652
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   653
(* standard version *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   654
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   655
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   656
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   657
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   658
by (auto dest: DERIV_chain simp add: o_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   659
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   660
(*derivative of linear multiplication*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   661
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   662
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   663
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   664
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   665
apply (cut_tac DERIV_power [OF DERIV_Id])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   666
apply (simp add: real_scaleR_def real_of_nat_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   667
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   668
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   669
(* NS version *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   670
lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   671
by (simp add: NSDERIV_DERIV_iff DERIV_pow)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   672
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   673
text{*Power of -1*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   674
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   675
lemma DERIV_inverse:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   676
  fixes x :: "'a::{real_normed_field,recpower}"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   677
  shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   678
by (drule DERIV_inverse' [OF DERIV_Id]) (simp add: power_Suc)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   679
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   680
text{*Derivative of inverse*}
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   681
lemma DERIV_inverse_fun:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   682
  fixes x :: "'a::{real_normed_field,recpower}"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   683
  shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   684
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   685
by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   686
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   687
lemma NSDERIV_inverse_fun:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   688
  fixes x :: "'a::{real_normed_field,recpower}"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   689
  shows "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   690
      ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   691
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   692
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   693
text{*Derivative of quotient*}
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   694
lemma DERIV_quotient:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   695
  fixes x :: "'a::{real_normed_field,recpower}"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   696
  shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   697
       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   698
by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   699
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   700
lemma NSDERIV_quotient:
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   701
  fixes x :: "'a::{real_normed_field,recpower}"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   702
  shows "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   703
       ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   704
                            - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   705
by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   706
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   707
lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   708
      \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   709
by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   710
                   mult_commute)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   711
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   712
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   713
by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   714
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   715
lemma CARAT_DERIVD:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   716
  assumes all: "\<forall>z. f z - f x = g z * (z-x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   717
      and nsc: "isNSCont g x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   718
  shows "NSDERIV f x :> g x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   719
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   720
  from nsc
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   721
  have "\<forall>w. w \<noteq> star_of x \<and> w \<approx> star_of x \<longrightarrow>
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   722
         ( *f* g) w * (w - star_of x) / (w - star_of x) \<approx>
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   723
         star_of (g x)"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   724
    by (simp add: isNSCont_def nonzero_mult_divide_cancel_right)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   725
  thus ?thesis using all
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   726
    by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   727
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   728
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   729
subsubsection {* Differentiability predicate *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   730
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   731
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   732
by (simp add: differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   733
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   734
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   735
by (force simp add: differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   736
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   737
lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   738
by (simp add: NSdifferentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   739
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   740
lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   741
by (force simp add: NSdifferentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   742
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   743
lemma differentiable_const: "(\<lambda>z. a) differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   744
  apply (unfold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   745
  apply (rule_tac x=0 in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   746
  apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   747
  done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   748
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   749
lemma differentiable_sum:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   750
  assumes "f differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   751
  and "g differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   752
  shows "(\<lambda>x. f x + g x) differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   753
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   754
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   755
  then obtain df where "DERIV f x :> df" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   756
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   757
  then obtain dg where "DERIV g x :> dg" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   758
  ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   759
  hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   760
  thus ?thesis by (fold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   761
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   762
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   763
lemma differentiable_diff:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   764
  assumes "f differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   765
  and "g differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   766
  shows "(\<lambda>x. f x - g x) differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   767
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   768
  from prems have "f differentiable x" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   769
  moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   770
  from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   771
  then obtain dg where "DERIV g x :> dg" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   772
  then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   773
  hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   774
  hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   775
  ultimately 
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   776
  show ?thesis
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   777
    by (auto simp: diff_def dest: differentiable_sum)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   778
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   779
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   780
lemma differentiable_mult:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   781
  assumes "f differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   782
  and "g differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   783
  shows "(\<lambda>x. f x * g x) differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   784
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   785
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   786
  then obtain df where "DERIV f x :> df" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   787
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   788
  then obtain dg where "DERIV g x :> dg" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   789
  ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   790
  hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   791
  thus ?thesis by (fold differentiable_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   792
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   793
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   794
subsection {*(NS) Increment*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   795
lemma incrementI:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   796
      "f NSdifferentiable x ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   797
      increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   798
      hypreal_of_real (f x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   799
by (simp add: increment_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   800
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   801
lemma incrementI2: "NSDERIV f x :> D ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   802
     increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   803
     hypreal_of_real (f x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   804
apply (erule NSdifferentiableI [THEN incrementI])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   805
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   806
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   807
(* The Increment theorem -- Keisler p. 65 *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   808
lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   809
      ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   810
apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   811
apply (drule bspec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   812
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   813
apply (frule_tac b1 = "hypreal_of_real (D) + y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   814
        in hypreal_mult_right_cancel [THEN iffD2])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   815
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)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   816
apply assumption
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   817
apply (simp add: times_divide_eq_right [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   818
apply (auto simp add: left_distrib)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   819
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   820
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   821
lemma increment_thm2:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   822
     "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   823
      ==> \<exists>e \<in> Infinitesimal. increment f x h =
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   824
              hypreal_of_real(D)*h + e*h"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   825
by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   826
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   827
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   828
lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   829
      ==> increment f x h \<approx> 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   830
apply (drule increment_thm2,
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   831
       auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   832
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   833
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   834
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   835
subsection {* Nested Intervals and Bisection *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   836
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   837
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   838
     All considerably tidied by lcp.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   839
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   840
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)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   841
apply (induct "no")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   842
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   843
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   844
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   845
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   846
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   847
         \<forall>n. f(n) \<le> g(n) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   848
      ==> Bseq (f :: nat \<Rightarrow> real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   849
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   850
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   851
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   852
apply (rule_tac y = "g (Suc na)" in order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   853
apply (induct_tac [2] "na")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   854
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   855
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   856
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   857
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   858
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   859
         \<forall>n. f(n) \<le> g(n) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   860
      ==> Bseq (g :: nat \<Rightarrow> real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   861
apply (subst Bseq_minus_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   862
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   863
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   864
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   865
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   866
lemma f_inc_imp_le_lim:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   867
  fixes f :: "nat \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   868
  shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   869
apply (rule linorder_not_less [THEN iffD1])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   870
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   871
apply (drule real_less_sum_gt_zero)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   872
apply (drule_tac x = "f n + - lim f" in spec, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   873
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   874
apply (subgoal_tac "lim f \<le> f (no + n) ")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   875
apply (drule_tac no=no and m=n in lemma_f_mono_add)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   876
apply (auto simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   877
apply (induct_tac "no")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   878
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   879
apply (auto intro: order_trans simp add: diff_minus abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   880
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   881
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   882
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   883
apply (rule LIMSEQ_minus [THEN limI])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   884
apply (simp add: convergent_LIMSEQ_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   885
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   886
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   887
lemma g_dec_imp_lim_le:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   888
  fixes g :: "nat \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   889
  shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   890
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   891
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   892
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   893
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   894
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   895
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   896
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   897
         \<forall>n. f(n) \<le> g(n) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   898
      ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   899
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   900
apply (subgoal_tac "monoseq f & monoseq g")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   901
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   902
apply (subgoal_tac "Bseq f & Bseq g")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   903
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   904
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   905
apply (rule_tac x = "lim f" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   906
apply (rule_tac x = "lim g" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   907
apply (auto intro: LIMSEQ_le)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   908
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   909
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   910
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   911
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   912
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   913
         \<forall>n. f(n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   914
         (%n. f(n) - g(n)) ----> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   915
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   916
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   917
apply (drule lemma_nest, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   918
apply (subgoal_tac "l = m")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   919
apply (drule_tac [2] X = f in LIMSEQ_diff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   920
apply (auto intro: LIMSEQ_unique)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   921
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   922
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   923
text{*The universal quantifiers below are required for the declaration
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   924
  of @{text Bolzano_nest_unique} below.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   925
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   926
lemma Bolzano_bisect_le:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   927
 "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   928
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   929
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   930
apply (auto simp add: Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   931
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   932
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   933
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   934
   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   935
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   936
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   937
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   938
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   939
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   940
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   941
   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   942
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   943
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   944
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   945
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   946
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   947
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   948
apply (auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   949
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   950
apply (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   951
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   952
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   953
lemma Bolzano_bisect_diff:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   954
     "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   955
      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   956
      (b-a) / (2 ^ n)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   957
apply (induct "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   958
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   959
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   960
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   961
lemmas Bolzano_nest_unique =
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   962
    lemma_nest_unique
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   963
    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   964
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   965
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   966
lemma not_P_Bolzano_bisect:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   967
  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   968
      and notP: "~ P(a,b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   969
      and le:   "a \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   970
  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   971
proof (induct n)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   972
  case 0 thus ?case by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   973
 next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   974
  case (Suc n)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   975
  thus ?case
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   976
 by (auto simp del: surjective_pairing [symmetric]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   977
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   978
     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   979
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   980
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   981
(*Now we re-package P_prem as a formula*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   982
lemma not_P_Bolzano_bisect':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   983
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   984
         ~ P(a,b);  a \<le> b |] ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   985
      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   986
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   987
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   988
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   989
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   990
lemma lemma_BOLZANO:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   991
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   992
         \<forall>x. \<exists>d::real. 0 < d &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   993
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   994
         a \<le> b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   995
      ==> P(a,b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   996
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   997
apply (rule LIMSEQ_minus_cancel)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   998
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   999
apply (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1000
apply (drule not_P_Bolzano_bisect', assumption+)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1001
apply (rename_tac "l")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1002
apply (drule_tac x = l in spec, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1003
apply (simp add: LIMSEQ_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1004
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1005
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1006
apply (drule real_less_half_sum, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1007
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1008
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1009
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1010
apply (simp_all (no_asm_simp))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1011
apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1012
apply (simp (no_asm_simp) add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1013
apply (rule real_sum_of_halves [THEN subst])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1014
apply (rule add_strict_mono)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1015
apply (simp_all add: diff_minus [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1016
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1017
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1018
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1019
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1020
       (\<forall>x. \<exists>d::real. 0 < d &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1021
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1022
      --> (\<forall>a b. a \<le> b --> P(a,b))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1023
apply clarify
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1024
apply (blast intro: lemma_BOLZANO)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1025
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1026
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1027
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1028
subsection {* Intermediate Value Theorem *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1029
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1030
text {*Prove Contrapositive by Bisection*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1031
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1032
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1033
         a \<le> b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1034
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1035
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1036
apply (rule contrapos_pp, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1037
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1038
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1039
apply simp_all
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1040
apply (simp add: isCont_iff LIM_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1041
apply (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1042
apply (subgoal_tac "a \<le> x & x \<le> b")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1043
 prefer 2
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1044
 apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1045
 apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1046
apply (drule_tac x = x in spec)+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1047
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1048
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1049
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1050
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1051
apply (drule_tac x = s in spec, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1052
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1053
apply (drule_tac x = "ba-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1054
apply (simp_all add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1055
apply (drule_tac x = "aa-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1056
apply (case_tac "x \<le> aa", simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1057
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1058
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1059
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1060
         a \<le> b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1061
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1062
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1063
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1064
apply (drule IVT [where f = "%x. - f x"], assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1065
apply (auto intro: isCont_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1066
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1067
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1068
(*HOL style here: object-level formulations*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1069
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1070
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1071
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1072
apply (blast intro: IVT)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1073
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1074
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1075
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1076
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1077
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1078
apply (blast intro: IVT2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1079
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1080
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1081
text{*By bisection, function continuous on closed interval is bounded above*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1082
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1083
lemma isCont_bounded:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1084
     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1085
      ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1086
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M)" in lemma_BOLZANO2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1087
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1088
apply simp_all
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1089
apply (rename_tac x xa ya M Ma)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1090
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1091
apply (rule_tac x = Ma in exI, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1092
apply (cut_tac x = xb and y = xa in linorder_linear, force)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1093
apply (rule_tac x = M in exI, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1094
apply (cut_tac x = xb and y = xa in linorder_linear, force)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1095
apply (case_tac "a \<le> x & x \<le> b")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1096
apply (rule_tac [2] x = 1 in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1097
prefer 2 apply force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1098
apply (simp add: LIM_def isCont_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1099
apply (drule_tac x = x in spec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1100
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1101
apply (drule_tac x = 1 in spec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1102
apply (rule_tac x = s in exI, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1103
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1104
apply (drule_tac x = "xa-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1105
apply (auto simp add: abs_ge_self)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1106
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1107
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1108
text{*Refine the above to existence of least upper bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1109
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1110
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1111
      (\<exists>t. isLub UNIV S t)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1112
by (blast intro: reals_complete)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1113
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1114
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1115
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1116
                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1117
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1118
        in lemma_reals_complete)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1119
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1120
apply (drule isCont_bounded, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1121
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1122
apply (rule exI, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1123
apply (auto dest!: spec simp add: linorder_not_less)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1124
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1125
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1126
text{*Now show that it attains its upper bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1127
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1128
lemma isCont_eq_Ub:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1129
  assumes le: "a \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1130
      and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1131
  shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1132
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1133
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1134
  from isCont_has_Ub [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1135
  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1136
             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1137
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1138
  proof (intro exI, intro conjI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1139
    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1140
    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1141
    proof (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1142
      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1143
      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1144
        by (fastsimp simp add: linorder_not_le [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1145
      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1146
        by (auto simp add: isCont_inverse isCont_diff con)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1147
      from isCont_bounded [OF le this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1148
      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1149
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1150
        by (simp add: M3 compare_rls)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1151
      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1152
        by (auto intro: order_le_less_trans [of _ k])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1153
      with Minv
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1154
      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1155
        by (intro strip less_imp_inverse_less, simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1156
      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1157
        by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1158
      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1159
        by (simp, arith)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1160
      from M2 [OF this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1161
      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1162
      thus False using invlt [of x] by force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1163
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1164
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1165
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1166
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1167
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1168
text{*Same theorem for lower bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1169
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1170
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1171
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1172
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1173
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1174
prefer 2 apply (blast intro: isCont_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1175
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1176
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1177
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1178
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1179
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1180
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1181
text{*Another version.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1182
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1183
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1184
      ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1185
          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1186
apply (frule isCont_eq_Lb)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1187
apply (frule_tac [2] isCont_eq_Ub)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1188
apply (assumption+, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1189
apply (rule_tac x = "f x" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1190
apply (rule_tac x = "f xa" in exI, simp, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1191
apply (cut_tac x = x and y = xa in linorder_linear, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1192
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1193
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1194
apply (rule_tac [2] x = xb in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1195
apply (rule_tac [4] x = xb in exI, simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1196
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1197
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1198
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1199
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1200
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1201
lemma DERIV_left_inc:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1202
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1203
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1204
      and l:   "0 < l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1205
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1206
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1207
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1208
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1209
    by (simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1210
  then obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1211
        where s:   "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1212
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1213
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1214
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1215
  proof (intro exI conjI strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1216
    show "0<s" .
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1217
    fix h::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1218
    assume "0 < h" "h < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1219
    with all [of h] show "f x < f (x+h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1220
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1221
    split add: split_if_asm)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1222
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1223
      with l
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1224
      have "0 < (f (x+h) - f x) / h" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1225
      thus "f x < f (x+h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1226
  by (simp add: pos_less_divide_eq h)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1227
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1228
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1229
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1230
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1231
lemma DERIV_left_dec:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1232
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1233
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1234
      and l:   "l < 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1235
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1236
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1237
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1238
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1239
    by (simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1240
  then obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1241
        where s:   "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1242
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1243
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1244
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1245
  proof (intro exI conjI strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1246
    show "0<s" .
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1247
    fix h::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1248
    assume "0 < h" "h < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1249
    with all [of "-h"] show "f x < f (x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1250
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1251
    split add: split_if_asm)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1252
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1253
      with l
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1254
      have "0 < (f (x-h) - f x) / h" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1255
      thus "f x < f (x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1256
  by (simp add: pos_less_divide_eq h)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1257
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1258
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1259
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1260
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1261
lemma DERIV_local_max:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1262
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1263
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1264
      and d:   "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1265
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1266
  shows "l = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1267
proof (cases rule: linorder_cases [of l 0])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1268
  case equal show ?thesis .
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1269
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1270
  case less
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1271
  from DERIV_left_dec [OF der less]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1272
  obtain d' where d': "0 < d'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1273
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1274
  from real_lbound_gt_zero [OF d d']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1275
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1276
  with lt le [THEN spec [where x="x-e"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1277
  show ?thesis by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1278
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1279
  case greater
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1280
  from DERIV_left_inc [OF der greater]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1281
  obtain d' where d': "0 < d'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1282
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1283
  from real_lbound_gt_zero [OF d d']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1284
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1285
  with lt le [THEN spec [where x="x+e"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1286
  show ?thesis by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1287
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1288
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1289
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1290
text{*Similar theorem for a local minimum*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1291
lemma DERIV_local_min:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1292
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1293
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1294
by (drule DERIV_minus [THEN DERIV_local_max], auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1295
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1296
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1297
text{*In particular, if a function is locally flat*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1298
lemma DERIV_local_const:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1299
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1300
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1301
by (auto dest!: DERIV_local_max)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1302
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1303
text{*Lemma about introducing open ball in open interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1304
lemma lemma_interval_lt:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1305
     "[| a < x;  x < b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1306
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1307
apply (simp add: abs_interval_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1308
apply (insert linorder_linear [of "x-a" "b-x"], safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1309
apply (rule_tac x = "x-a" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1310
apply (rule_tac [2] x = "b-x" in exI, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1311
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1312
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1313
lemma lemma_interval: "[| a < x;  x < b |] ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1314
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1315
apply (drule lemma_interval_lt, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1316
apply (auto intro!: exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1317
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1318
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1319
text{*Rolle's Theorem.
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1320
   If @{term f} is defined and continuous on the closed interval
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1321
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1322
   and @{term "f(a) = f(b)"},
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1323
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1324
theorem Rolle:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1325
  assumes lt: "a < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1326
      and eq: "f(a) = f(b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1327
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1328
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1329
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1330
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1331
  have le: "a \<le> b" using lt by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1332
  from isCont_eq_Ub [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1333
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1334
             and alex: "a \<le> x" and xleb: "x \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1335
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1336
  from isCont_eq_Lb [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1337
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1338
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1339
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1340
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1341
  proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1342
    assume axb: "a < x & x < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1343
        --{*@{term f} attains its maximum within the interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1344
    hence ax: "a<x" and xb: "x<b" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1345
    from lemma_interval [OF ax xb]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1346
    obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1347
      by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1348
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1349
      by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1350
    from differentiableD [OF dif [OF axb]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1351
    obtain l where der: "DERIV f x :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1352
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1353
        --{*the derivative at a local maximum is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1354
    thus ?thesis using ax xb der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1355
  next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1356
    assume notaxb: "~ (a < x & x < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1357
    hence xeqab: "x=a | x=b" using alex xleb by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1358
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1359
    show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1360
    proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1361
      assume ax'b: "a < x' & x' < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1362
        --{*@{term f} attains its minimum within the interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1363
      hence ax': "a<x'" and x'b: "x'<b" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1364
      from lemma_interval [OF ax' x'b]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1365
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1366
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1367
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1368
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1369
      from differentiableD [OF dif [OF ax'b]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1370
      obtain l where der: "DERIV f x' :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1371
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1372
        --{*the derivative at a local minimum is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1373
      thus ?thesis using ax' x'b der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1374
    next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1375
      assume notax'b: "~ (a < x' & x' < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1376
        --{*@{term f} is constant througout the interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1377
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1378
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1379
      from dense [OF lt]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1380
      obtain r where ar: "a < r" and rb: "r < b" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1381
      from lemma_interval [OF ar rb]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1382
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1383
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1384
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1385
      proof (clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1386
        fix z::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1387
        assume az: "a \<le> z" and zb: "z \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1388
        show "f z = f b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1389
        proof (rule order_antisym)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1390
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1391
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1392
        qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1393
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1394
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1395
      proof (intro strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1396
        fix y::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1397
        assume lt: "\<bar>r-y\<bar> < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1398
        hence "f y = f b" by (simp add: eq_fb bound)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1399
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1400
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1401
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1402
      obtain l where der: "DERIV f r :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1403
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1404
        --{*the derivative of a constant function is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1405
      thus ?thesis using ar rb der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1406
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1407
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1408
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1409
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1410
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1411
subsection{*Mean Value Theorem*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1412
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1413
lemma lemma_MVT:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1414
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1415
proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1416
  assume "a=b" thus ?thesis by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1417
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1418
  assume "a\<noteq>b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1419
  hence ba: "b-a \<noteq> 0" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1420
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1421
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset