src/HOL/Deriv.thy
author paulson
Fri, 13 Nov 2009 11:33:33 +0000
changeset 33654 abf780db30ea
parent 31902 862ae16a799d
child 33659 2d7ab9458518
permissions -rw-r--r--
A number of theorems contributed by Jeremy Avigad
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(*  Title       : Deriv.thy
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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 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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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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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_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
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by (simp add: deriv_def 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_ident)
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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)
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  thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
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    by (simp add: LIM_def dist_norm)
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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_distribs)
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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_mult 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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text{*Differentiation of finite sum*}
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lemma DERIV_setsum:
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  assumes "finite S"
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  and "\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x :> (f' x n)"
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  shows "DERIV (%x. setsum (f x) S) x :> setsum (f' x) S"
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  using assms by induct (auto intro!: DERIV_add)
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lemma DERIV_sumr [rule_format (no_asm)]:
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     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
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      --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
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  by (auto intro: DERIV_setsum)
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text{*Alternative definition for differentiability*}
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lemma DERIV_LIM_iff:
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  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
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     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
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      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
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apply (rule iffI)
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apply (drule_tac k="- a" in LIM_offset)
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apply (simp add: diff_minus)
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apply (drule_tac k="a" in LIM_offset)
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apply (simp add: add_commute)
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done
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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
   143
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
   144
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   145
lemma inverse_diff_inverse:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   146
  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   147
   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29472
diff changeset
   148
by (simp add: algebra_simps)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   149
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   150
lemma DERIV_inverse_lemma:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   151
  "\<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
   152
   \<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
   153
     = - (inverse a * ((a - b) / h) * inverse b)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   154
by (simp add: inverse_diff_inverse)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   155
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   156
lemma DERIV_inverse':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   157
  assumes der: "DERIV f x :> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   158
  assumes neq: "f x \<noteq> 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   159
  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
   160
    (is "DERIV _ _ :> ?E")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   161
proof (unfold DERIV_iff2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   162
  from der have lim_f: "f -- x --> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   163
    by (rule DERIV_isCont [unfolded isCont_def])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   165
  from neq have "0 < norm (f x)" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   166
  with LIM_D [OF lim_f] obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   167
    where s: "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   168
    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
   169
                  \<Longrightarrow> norm (f z - f x) < norm (f x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   170
    by fast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   171
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   172
  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
   173
  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
   174
    fix z
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   175
    assume "z \<noteq> x" "norm (z - x) < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   176
    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
   177
    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
   178
    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
   179
          - (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
   180
      using neq by (rule DERIV_inverse_lemma)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   181
  next
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   182
    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
   183
      by (unfold DERIV_iff2)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   184
    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
   185
          -- x --> ?E"
22613
2f119f54d150 remove redundant lemmas
huffman
parents: 21810
diff changeset
   186
      by (intro LIM_mult LIM_inverse LIM_minus LIM_const lim_f neq)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   187
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   188
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   189
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   190
lemma DERIV_divide:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   191
  "\<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
   192
   \<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
   193
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
   194
          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
   195
apply (erule subst)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   196
apply (unfold divide_inverse)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   197
apply (erule DERIV_mult')
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   198
apply (erule (1) DERIV_inverse')
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23441
diff changeset
   199
apply (simp add: ring_distribs nonzero_inverse_mult_distrib)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   200
apply (simp add: mult_ac)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   201
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   202
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   203
lemma DERIV_power_Suc:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   204
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   205
  assumes f: "DERIV f x :> D"
23431
25ca91279a9b change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents: 23413
diff changeset
   206
  shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   207
proof (induct n)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   208
case 0
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   209
  show ?case by (simp add: f)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   210
case (Suc k)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   211
  from DERIV_mult' [OF f Suc] show ?case
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23441
diff changeset
   212
    apply (simp only: of_nat_Suc ring_distribs mult_1_left)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29472
diff changeset
   213
    apply (simp only: power_Suc algebra_simps)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   214
    done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   215
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   216
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   217
lemma DERIV_power:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   218
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   219
  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
   220
  shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   221
by (cases "n", simp, simp add: DERIV_power_Suc f del: power_Suc)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   222
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   223
text {* Caratheodory formulation of derivative at a point *}
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   224
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   225
lemma CARAT_DERIV:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   226
     "(DERIV f x :> l) =
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   227
      (\<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
   228
      (is "?lhs = ?rhs")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   229
proof
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   230
  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
   231
  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
   232
  proof (intro exI conjI)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   233
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23412
diff changeset
   234
    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   235
    show "isCont ?g x" using der
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   236
      by (simp add: isCont_iff DERIV_iff diff_minus
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   237
               cong: LIM_equal [rule_format])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   238
    show "?g x = l" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   239
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   240
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   241
  assume "?rhs"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   242
  then obtain g where
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   243
    "(\<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
   244
  thus "(DERIV f x :> l)"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23412
diff changeset
   245
     by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   246
qed
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
lemma DERIV_chain':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   249
  assumes f: "DERIV f x :> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   250
  assumes g: "DERIV g (f x) :> E"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   251
  shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   252
proof (unfold DERIV_iff2)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   253
  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
huffman
parents:
diff changeset
   254
    and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   255
    using CARAT_DERIV [THEN iffD1, OF g] by fast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   256
  from f have "f -- x --> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   257
    by (rule DERIV_isCont [unfolded isCont_def])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   258
  with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21199
diff changeset
   259
    by (rule isCont_LIM_compose)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   260
  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
huffman
parents: 21404
diff changeset
   261
          -- x --> d (f x) * D"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   262
    by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   263
  thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   264
    by (simp add: d dfx real_scaleR_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   265
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   266
31899
1a7ade46061b fixed document (DERIV_intros);
wenzelm
parents: 31880
diff changeset
   267
text {*
1a7ade46061b fixed document (DERIV_intros);
wenzelm
parents: 31880
diff changeset
   268
 Let's do the standard proof, though theorem
1a7ade46061b fixed document (DERIV_intros);
wenzelm
parents: 31880
diff changeset
   269
 @{text "LIM_mult2"} follows from a NS proof
1a7ade46061b fixed document (DERIV_intros);
wenzelm
parents: 31880
diff changeset
   270
*}
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   271
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   272
lemma DERIV_cmult:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   273
      "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
   274
by (drule DERIV_mult' [OF DERIV_const], simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   275
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   276
lemma DERIV_cdivide: "DERIV f x :> D ==> DERIV (%x. f x / c) x :> D / c"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   277
  apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x :> (1 / c) * D", force)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   278
  apply (erule DERIV_cmult)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   279
  done
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text {* Standard version *}
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lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
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by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
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lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
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by (auto dest: DERIV_chain simp add: o_def)
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text {* Derivative of linear multiplication *}
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lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
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by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
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lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
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apply (cut_tac DERIV_power [OF DERIV_ident])
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apply (simp add: real_scaleR_def real_of_nat_def)
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done
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text {* Power of @{text "-1"} *}
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lemma DERIV_inverse:
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  fixes x :: "'a::{real_normed_field}"
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  shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
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by (drule DERIV_inverse' [OF DERIV_ident]) simp
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text {* Derivative of inverse *}
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lemma DERIV_inverse_fun:
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  fixes x :: "'a::{real_normed_field}"
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  shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
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      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
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by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
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text {* Derivative of quotient *}
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lemma DERIV_quotient:
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  fixes x :: "'a::{real_normed_field}"
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  shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
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       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
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by (drule (2) DERIV_divide) (simp add: mult_commute)
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lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
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by auto
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text {* @{text "DERIV_intros"} *}
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ML {*
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structure Deriv_Intros = Named_Thms
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(
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  val name = "DERIV_intros"
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  val description = "DERIV introduction rules"
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)
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*}
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setup Deriv_Intros.setup
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lemma DERIV_cong: "\<lbrakk> DERIV f x :> X ; X = Y \<rbrakk> \<Longrightarrow> DERIV f x :> Y"
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  by simp
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declare
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  DERIV_const[THEN DERIV_cong, DERIV_intros]
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  DERIV_ident[THEN DERIV_cong, DERIV_intros]
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  DERIV_add[THEN DERIV_cong, DERIV_intros]
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  DERIV_minus[THEN DERIV_cong, DERIV_intros]
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  DERIV_mult[THEN DERIV_cong, DERIV_intros]
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  DERIV_diff[THEN DERIV_cong, DERIV_intros]
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  DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
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  DERIV_divide[THEN DERIV_cong, DERIV_intros]
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  DERIV_power[where 'a=real, THEN DERIV_cong,
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              unfolded real_of_nat_def[symmetric], DERIV_intros]
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  DERIV_setsum[THEN DERIV_cong, DERIV_intros]
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subsection {* Differentiability predicate *}
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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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   355
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lemma differentiableE [elim?]:
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   357
  assumes "f differentiable x"
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   358
  obtains df where "DERIV f x :> df"
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   359
  using prems unfolding differentiable_def ..
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   360
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lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
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   362
by (simp add: differentiable_def)
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   363
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
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   364
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
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parents:
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   365
by (force simp add: differentiable_def)
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   366
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   367
lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
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   368
  by (rule DERIV_ident [THEN differentiableI])
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   369
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   370
lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
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   371
  by (rule DERIV_const [THEN differentiableI])
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parents:
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   372
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   373
lemma differentiable_compose:
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   374
  assumes f: "f differentiable (g x)"
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   375
  assumes g: "g differentiable x"
6a5f1d8d7344 more proofs about differentiable
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   376
  shows "(\<lambda>x. f (g x)) differentiable x"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   377
proof -
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diff changeset
   378
  from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
6a5f1d8d7344 more proofs about differentiable
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   379
  moreover
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   380
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
6a5f1d8d7344 more proofs about differentiable
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   381
  ultimately
6a5f1d8d7344 more proofs about differentiable
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diff changeset
   382
  have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   383
  thus ?thesis by (rule differentiableI)
6a5f1d8d7344 more proofs about differentiable
huffman
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diff changeset
   384
qed
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diff changeset
   385
6a5f1d8d7344 more proofs about differentiable
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diff changeset
   386
lemma differentiable_sum [simp]:
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   387
  assumes "f differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   388
  and "g differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   389
  shows "(\<lambda>x. f x + g x) differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   390
proof -
29169
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   391
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   392
  moreover
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   393
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   394
  ultimately
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   395
  have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   396
  thus ?thesis by (rule differentiableI)
6a5f1d8d7344 more proofs about differentiable
huffman
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diff changeset
   397
qed
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parents: 29166
diff changeset
   398
6a5f1d8d7344 more proofs about differentiable
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   399
lemma differentiable_minus [simp]:
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   400
  assumes "f differentiable x"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   401
  shows "(\<lambda>x. - f x) differentiable x"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   402
proof -
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huffman
parents: 29166
diff changeset
   403
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   404
  hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
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huffman
parents: 29166
diff changeset
   405
  thus ?thesis by (rule differentiableI)
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huffman
parents:
diff changeset
   406
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   407
29169
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diff changeset
   408
lemma differentiable_diff [simp]:
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0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   409
  assumes "f differentiable x"
29169
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   410
  assumes "g differentiable x"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   411
  shows "(\<lambda>x. f x - g x) differentiable x"
29169
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huffman
parents: 29166
diff changeset
   412
  unfolding diff_minus using prems by simp
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   413
6a5f1d8d7344 more proofs about differentiable
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diff changeset
   414
lemma differentiable_mult [simp]:
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diff changeset
   415
  assumes "f differentiable x"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   416
  assumes "g differentiable x"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   417
  shows "(\<lambda>x. f x * g x) differentiable x"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   418
proof -
29169
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parents: 29166
diff changeset
   419
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   420
  moreover
29169
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huffman
parents: 29166
diff changeset
   421
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   422
  ultimately
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   423
  have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   424
  thus ?thesis by (rule differentiableI)
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huffman
parents:
diff changeset
   425
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   426
29169
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huffman
parents: 29166
diff changeset
   427
lemma differentiable_inverse [simp]:
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parents: 29166
diff changeset
   428
  assumes "f differentiable x" and "f x \<noteq> 0"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   429
  shows "(\<lambda>x. inverse (f x)) differentiable x"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   430
proof -
29169
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huffman
parents: 29166
diff changeset
   431
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   432
  hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   433
    using `f x \<noteq> 0` by (rule DERIV_inverse')
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   434
  thus ?thesis by (rule differentiableI)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   435
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   436
29169
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   437
lemma differentiable_divide [simp]:
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   438
  assumes "f differentiable x"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   439
  assumes "g differentiable x" and "g x \<noteq> 0"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   440
  shows "(\<lambda>x. f x / g x) differentiable x"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   441
  unfolding divide_inverse using prems by simp
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   442
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   443
lemma differentiable_power [simp]:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30273
diff changeset
   444
  fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a"
29169
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   445
  assumes "f differentiable x"
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   446
  shows "(\<lambda>x. f x ^ n) differentiable x"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   447
  by (induct n, simp, simp add: prems)
29169
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   448
22984
67d434ad9ef8 section labels
huffman
parents: 22653
diff changeset
   449
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   450
subsection {* Nested Intervals and Bisection *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   451
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   452
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   453
     All considerably tidied by lcp.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   454
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   455
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
   456
apply (induct "no")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   457
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   458
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   459
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   460
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
   461
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   462
         \<forall>n. f(n) \<le> g(n) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   463
      ==> Bseq (f :: nat \<Rightarrow> real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   464
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
   465
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   466
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   467
apply (rule_tac y = "g (Suc na)" in order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   468
apply (induct_tac [2] "na")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   469
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   470
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   471
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   472
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
   473
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   474
         \<forall>n. f(n) \<le> g(n) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   475
      ==> Bseq (g :: nat \<Rightarrow> real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   476
apply (subst Bseq_minus_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   477
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
   478
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   479
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   480
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   481
lemma f_inc_imp_le_lim:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   482
  fixes f :: "nat \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   483
  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
   484
apply (rule linorder_not_less [THEN iffD1])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   485
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   486
apply (drule real_less_sum_gt_zero)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   487
apply (drule_tac x = "f n + - lim f" in spec, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   488
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
   489
apply (subgoal_tac "lim f \<le> f (no + n) ")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   490
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
   491
apply (auto simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   492
apply (induct_tac "no")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   493
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   494
apply (auto intro: order_trans simp add: diff_minus abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   495
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   496
31404
05d2eddc5d41 generalize type of constant lim
huffman
parents: 31338
diff changeset
   497
lemma lim_uminus:
05d2eddc5d41 generalize type of constant lim
huffman
parents: 31338
diff changeset
   498
  fixes g :: "nat \<Rightarrow> 'a::real_normed_vector"
05d2eddc5d41 generalize type of constant lim
huffman
parents: 31338
diff changeset
   499
  shows "convergent g ==> lim (%x. - g x) = - (lim g)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   500
apply (rule LIMSEQ_minus [THEN limI])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   501
apply (simp add: convergent_LIMSEQ_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   502
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   503
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   504
lemma g_dec_imp_lim_le:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   505
  fixes g :: "nat \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   506
  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
   507
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   508
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
   509
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   510
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   511
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   512
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   513
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   514
         \<forall>n. f(n) \<le> g(n) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   515
      ==> \<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
   516
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   517
apply (subgoal_tac "monoseq f & monoseq g")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   518
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   519
apply (subgoal_tac "Bseq f & Bseq g")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   520
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
   521
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   522
apply (rule_tac x = "lim f" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   523
apply (rule_tac x = "lim g" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   524
apply (auto intro: LIMSEQ_le)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   525
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
   526
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   527
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   528
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
   529
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   530
         \<forall>n. f(n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   531
         (%n. f(n) - g(n)) ----> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   532
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   533
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   534
apply (drule lemma_nest, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   535
apply (subgoal_tac "l = m")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   536
apply (drule_tac [2] X = f in LIMSEQ_diff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   537
apply (auto intro: LIMSEQ_unique)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   538
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   539
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   540
text{*The universal quantifiers below are required for the declaration
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   541
  of @{text Bolzano_nest_unique} below.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   542
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   543
lemma Bolzano_bisect_le:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   544
 "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
   545
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   546
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   547
apply (auto simp add: Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   548
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   549
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   550
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   551
   \<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
   552
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   553
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   554
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   555
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   556
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   557
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   558
   \<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
   559
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   560
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   561
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   562
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   563
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   564
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
   565
apply (auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   566
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   567
apply (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   568
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   569
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   570
lemma Bolzano_bisect_diff:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   571
     "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   572
      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
   573
      (b-a) / (2 ^ n)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   574
apply (induct "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   575
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
   576
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   577
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   578
lemmas Bolzano_nest_unique =
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   579
    lemma_nest_unique
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   580
    [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
   581
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   582
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   583
lemma not_P_Bolzano_bisect:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   584
  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
   585
      and notP: "~ P(a,b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   586
      and le:   "a \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   587
  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
   588
proof (induct n)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   589
  case 0 show ?case using notP by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   590
 next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   591
  case (Suc n)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   592
  thus ?case
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   593
 by (auto simp del: surjective_pairing [symmetric]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   594
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   595
     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
   596
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   597
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   598
(*Now we re-package P_prem as a formula*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   599
lemma not_P_Bolzano_bisect':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   600
     "[| \<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
   601
         ~ P(a,b);  a \<le> b |] ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   602
      \<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
   603
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   604
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
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   607
lemma lemma_BOLZANO:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   608
     "[| \<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
   609
         \<forall>x. \<exists>d::real. 0 < d &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   610
                (\<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
   611
         a \<le> b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   612
      ==> P(a,b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   613
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   614
apply (rule LIMSEQ_minus_cancel)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   615
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
   616
apply (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   617
apply (drule not_P_Bolzano_bisect', assumption+)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   618
apply (rename_tac "l")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   619
apply (drule_tac x = l in spec, clarify)
31336
e17f13cd1280 generalize constants in SEQ.thy to class metric_space
huffman
parents: 31017
diff changeset
   620
apply (simp add: LIMSEQ_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   621
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
   622
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
   623
apply (drule real_less_half_sum, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   624
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
   625
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
   626
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   627
apply (simp_all (no_asm_simp))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   628
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
   629
apply (simp (no_asm_simp) add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   630
apply (rule real_sum_of_halves [THEN subst])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   631
apply (rule add_strict_mono)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   632
apply (simp_all add: diff_minus [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   633
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   634
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   635
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   636
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
   637
       (\<forall>x. \<exists>d::real. 0 < d &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   638
                (\<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
   639
      --> (\<forall>a b. a \<le> b --> P(a,b))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   640
apply clarify
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   641
apply (blast intro: lemma_BOLZANO)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   642
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   643
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   644
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   645
subsection {* Intermediate Value Theorem *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   646
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   647
text {*Prove Contrapositive by Bisection*}
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 IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   650
         a \<le> b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   651
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   652
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   653
apply (rule contrapos_pp, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   654
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
   655
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   656
apply simp_all
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   657
apply (simp add: isCont_iff LIM_eq)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   658
apply (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   659
apply (subgoal_tac "a \<le> x & x \<le> b")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   660
 prefer 2
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   661
 apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   662
 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
   663
apply (drule_tac x = x in spec)+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   664
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   665
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
   666
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   667
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   668
apply (drule_tac x = s in spec, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   669
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
   670
apply (drule_tac x = "ba-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   671
apply (simp_all add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   672
apply (drule_tac x = "aa-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   673
apply (case_tac "x \<le> aa", simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   674
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   675
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   676
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
   677
         a \<le> b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   678
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   679
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   680
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
   681
apply (drule IVT [where f = "%x. - f x"], assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   682
apply (auto intro: isCont_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   683
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   684
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   685
(*HOL style here: object-level formulations*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   686
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
   687
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   688
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   689
apply (blast intro: IVT)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   690
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   691
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   692
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
   693
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   694
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   695
apply (blast intro: IVT2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   696
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   697
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   698
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   699
subsection {* Boundedness of continuous functions *}
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   700
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   701
text{*By bisection, function continuous on closed interval is bounded above*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   702
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   703
lemma isCont_bounded:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   704
     "[| 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
   705
      ==> \<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
   706
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
   707
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   708
apply simp_all
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   709
apply (rename_tac x xa ya M Ma)
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   710
apply (metis linorder_not_less order_le_less real_le_trans)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   711
apply (case_tac "a \<le> x & x \<le> b")
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   712
 prefer 2
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   713
 apply (rule_tac x = 1 in exI, force)
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   714
apply (simp add: LIM_eq isCont_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   715
apply (drule_tac x = x in spec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   716
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
   717
apply (drule_tac x = 1 in spec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   718
apply (rule_tac x = s in exI, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   719
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
   720
apply (drule_tac x = "xa-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   721
apply (auto simp add: abs_ge_self)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   722
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   723
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   724
text{*Refine the above to existence of least upper bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   725
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   726
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
   727
      (\<exists>t. isLub UNIV S t)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   728
by (blast intro: reals_complete)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   729
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   730
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
   731
         ==> \<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
   732
                   (\<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
   733
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
   734
        in lemma_reals_complete)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   735
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   736
apply (drule isCont_bounded, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   737
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
   738
apply (rule exI, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   739
apply (auto dest!: spec simp add: linorder_not_less)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   740
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   741
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   742
text{*Now show that it attains its upper bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   743
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   744
lemma isCont_eq_Ub:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   745
  assumes le: "a \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   746
      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
   747
  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
   748
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   749
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   750
  from isCont_has_Ub [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   751
  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
   752
             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
   753
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   754
  proof (intro exI, intro conjI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   755
    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
   756
    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
   757
    proof (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   758
      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
   759
      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
   760
        by (fastsimp simp add: linorder_not_le [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   761
      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
   762
        by (auto simp add: isCont_inverse isCont_diff con)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   763
      from isCont_bounded [OF le this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   764
      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
   765
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29472
diff changeset
   766
        by (simp add: M3 algebra_simps)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   767
      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
   768
        by (auto intro: order_le_less_trans [of _ k])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   769
      with Minv
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   770
      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
   771
        by (intro strip less_imp_inverse_less, simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   772
      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
   773
        by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   774
      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
   775
        by (simp, arith)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   776
      from M2 [OF this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   777
      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
   778
      thus False using invlt [of x] by force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   779
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   780
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   781
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   782
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   783
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   784
text{*Same theorem for lower bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   785
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   786
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
   787
         ==> \<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
   788
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   789
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
   790
prefer 2 apply (blast intro: isCont_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   791
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   792
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   793
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   794
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   795
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   796
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   797
text{*Another version.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   798
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   799
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
   800
      ==> \<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
   801
          (\<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
   802
apply (frule isCont_eq_Lb)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   803
apply (frule_tac [2] isCont_eq_Ub)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   804
apply (assumption+, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   805
apply (rule_tac x = "f x" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   806
apply (rule_tac x = "f xa" in exI, simp, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   807
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
   808
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
   809
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
   810
apply (rule_tac [2] x = xb in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   811
apply (rule_tac [4] x = xb in exI, simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   812
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   813
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   814
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   815
subsection {* Local extrema *}
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   816
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   817
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
   818
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   819
lemma DERIV_pos_inc_right:
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   820
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   821
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   822
      and l:   "0 < l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   823
  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
   824
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   825
  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
   826
  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
   827
    by (simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   828
  then obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   829
        where s:   "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   830
          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
   831
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   832
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   833
  proof (intro exI conjI strip)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   834
    show "0<s" using s .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   835
    fix h::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   836
    assume "0 < h" "h < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   837
    with all [of h] show "f x < f (x+h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   838
    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
   839
    split add: split_if_asm)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   840
      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
   841
      with l
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   842
      have "0 < (f (x+h) - f x) / h" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   843
      thus "f x < f (x+h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   844
  by (simp add: pos_less_divide_eq h)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   845
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   846
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   847
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   848
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   849
lemma DERIV_neg_dec_left:
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   850
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   851
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   852
      and l:   "l < 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   853
  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
   854
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   855
  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
   856
  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
   857
    by (simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   858
  then obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   859
        where s:   "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   860
          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
   861
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   862
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   863
  proof (intro exI conjI strip)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   864
    show "0<s" using s .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   865
    fix h::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   866
    assume "0 < h" "h < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   867
    with all [of "-h"] show "f x < f (x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   868
    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
   869
    split add: split_if_asm)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   870
      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
   871
      with l
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   872
      have "0 < (f (x-h) - f x) / h" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   873
      thus "f x < f (x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   874
  by (simp add: pos_less_divide_eq h)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   875
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   876
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   877
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   878
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   879
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   880
lemma DERIV_pos_inc_left:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   881
  fixes f :: "real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   882
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   883
  apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   884
  apply (auto simp add: DERIV_minus) 
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   885
  done
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   886
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   887
lemma DERIV_neg_dec_right:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   888
  fixes f :: "real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   889
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   890
  apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   891
  apply (auto simp add: DERIV_minus) 
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   892
  done
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   893
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   894
lemma DERIV_local_max:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   895
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   896
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   897
      and d:   "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   898
      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
   899
  shows "l = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   900
proof (cases rule: linorder_cases [of l 0])
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   901
  case equal thus ?thesis .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   902
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   903
  case less
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   904
  from DERIV_neg_dec_left [OF der less]
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   905
  obtain d' where d': "0 < d'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   906
             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
   907
  from real_lbound_gt_zero [OF d d']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   908
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   909
  with lt le [THEN spec [where x="x-e"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   910
  show ?thesis by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   911
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   912
  case greater
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   913
  from DERIV_pos_inc_right [OF der greater]
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   914
  obtain d' where d': "0 < d'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   915
             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
   916
  from real_lbound_gt_zero [OF d d']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   917
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   918
  with lt le [THEN spec [where x="x+e"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   919
  show ?thesis by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   920
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   921
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{*Similar theorem for a local minimum*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   924
lemma DERIV_local_min:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   925
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   926
  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
   927
by (drule DERIV_minus [THEN DERIV_local_max], auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   928
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   929
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   930
text{*In particular, if a function is locally flat*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   931
lemma DERIV_local_const:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   932
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   933
  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
   934
by (auto dest!: DERIV_local_max)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   935
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   936
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   937
subsection {* Rolle's Theorem *}
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   938
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   939
text{*Lemma about introducing open ball in open interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   940
lemma lemma_interval_lt:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   941
     "[| a < x;  x < b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   942
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
   943
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22984
diff changeset
   944
apply (simp add: abs_less_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   945
apply (insert linorder_linear [of "x-a" "b-x"], safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   946
apply (rule_tac x = "x-a" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   947
apply (rule_tac [2] x = "b-x" in exI, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   948
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   949
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   950
lemma lemma_interval: "[| a < x;  x < b |] ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   951
        \<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
   952
apply (drule lemma_interval_lt, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   953
apply (auto intro!: exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   954
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   955
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   956
text{*Rolle's Theorem.
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   957
   If @{term f} is defined and continuous on the closed interval
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   958
   @{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
   959
   and @{term "f(a) = f(b)"},
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   960
   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
   961
theorem Rolle:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   962
  assumes lt: "a < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   963
      and eq: "f(a) = f(b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   964
      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
   965
      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
   966
  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
   967
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   968
  have le: "a \<le> b" using lt by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   969
  from isCont_eq_Ub [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   970
  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
   971
             and alex: "a \<le> x" and xleb: "x \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   972
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   973
  from isCont_eq_Lb [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   974
  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
   975
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   976
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   977
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   978
  proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   979
    assume axb: "a < x & x < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   980
        --{*@{term f} attains its maximum within the interval*}
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
   981
    hence ax: "a<x" and xb: "x<b" by arith + 
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   982
    from lemma_interval [OF ax xb]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   983
    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
   984
      by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   985
    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
   986
      by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   987
    from differentiableD [OF dif [OF axb]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   988
    obtain l where der: "DERIV f x :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   989
    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
   990
        --{*the derivative at a local maximum is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   991
    thus ?thesis using ax xb der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   992
  next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   993
    assume notaxb: "~ (a < x & x < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   994
    hence xeqab: "x=a | x=b" using alex xleb by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   995
    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
   996
    show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   997
    proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   998
      assume ax'b: "a < x' & x' < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   999
        --{*@{term f} attains its minimum within the interval*}
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
  1000
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1001
      from lemma_interval [OF ax' x'b]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1002
      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
  1003
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1004
      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
  1005
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1006
      from differentiableD [OF dif [OF ax'b]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1007
      obtain l where der: "DERIV f x' :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1008
      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
  1009
        --{*the derivative at a local minimum is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1010
      thus ?thesis using ax' x'b der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1011
    next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1012
      assume notax'b: "~ (a < x' & x' < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1013
        --{*@{term f} is constant througout the interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1014
      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
  1015
      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
  1016
      from dense [OF lt]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1017
      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
  1018
      from lemma_interval [OF ar rb]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1019
      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
  1020
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1021
      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
  1022
      proof (clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1023
        fix z::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1024
        assume az: "a \<le> z" and zb: "z \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1025
        show "f z = f b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1026
        proof (rule order_antisym)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1027
          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
  1028
          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
  1029
        qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1030
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1031
      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
  1032
      proof (intro strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1033
        fix y::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1034
        assume lt: "\<bar>r-y\<bar> < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1035
        hence "f y = f b" by (simp add: eq_fb bound)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1036
        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
  1037
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1038
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1039
      obtain l where der: "DERIV f r :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1040
      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
  1041
        --{*the derivative of a constant function is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1042
      thus ?thesis using ar rb der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1043
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1044
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1045
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1046
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1047
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1048
subsection{*Mean Value Theorem*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1049
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1050
lemma lemma_MVT:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1051
     "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
  1052
proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1053
  assume "a=b" thus ?thesis by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1054
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1055
  assume "a\<noteq>b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1056
  hence ba: "b-a \<noteq> 0" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1057
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1058
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1059
        simp add: right_diff_distrib,
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1060
        simp add: left_diff_distrib)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1061
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1062
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1063
theorem MVT:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1064
  assumes lt:  "a < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1065
      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
  1066
      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
  1067
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1068
                   (f(b) - f(a) = (b-a) * l)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1069
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1070
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1071
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
23069
cdfff0241c12 rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents: 23044
diff changeset
  1072
    by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1073
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1074
  proof (clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1075
    fix x::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1076
    assume ax: "a < x" and xb: "x < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1077
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1078
    obtain l where der: "DERIV f x :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1079
    show "?F differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1080
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1081
          blast intro: DERIV_diff DERIV_cmult_Id der)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1082
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1083
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1084
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1085
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1086
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1087
    by (rule DERIV_cmult_Id)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1088
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1089
                   :> 0 + (f b - f a) / (b - a)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1090
    by (rule DERIV_add [OF der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1091
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1092
  proof (intro exI conjI)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1093
    show "a < z" using az .
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1094
    show "z < b" using zb .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1095
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1096
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1097
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1098
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1099
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1100
lemma MVT2:
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1101
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1102
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1103
apply (drule MVT)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1104
apply (blast intro: DERIV_isCont)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1105
apply (force dest: order_less_imp_le simp add: differentiable_def)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1106
apply (blast dest: DERIV_unique order_less_imp_le)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1107
done
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1108
21164
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
text{*A function is constant if its derivative is 0 over an interval.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1111
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1112
lemma DERIV_isconst_end:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1113
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1114
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1115
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1116
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1117
        ==> f b = f a"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1118
apply (drule MVT, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1119
apply (blast intro: differentiableI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1120
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1121
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1122
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1123
lemma DERIV_isconst1:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1124
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1125
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1126
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1127
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1128
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1129
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1130
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1131
apply (drule_tac b = x in DERIV_isconst_end, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1132
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1133
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1134
lemma DERIV_isconst2:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1135
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1136
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1137
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1138
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1139
         a \<le> x; x \<le> b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1140
        ==> f x = f a"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1141
apply (blast dest: DERIV_isconst1)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1142
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1143
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1144
lemma DERIV_isconst3: fixes a b x y :: real
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1145
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1146
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1147
  shows "f x = f y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1148
proof (cases "x = y")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1149
  case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1150
  let ?a = "min x y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1151
  let ?b = "max x y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1152
  
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1153
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1154
  proof (rule allI, rule impI)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1155
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1156
    hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1157
    hence "z \<in> {a<..<b}" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1158
    thus "DERIV f z :> 0" by (rule derivable)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1159
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1160
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1161
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1162
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1163
  have "?a < ?b" using `x \<noteq> y` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1164
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1165
  show ?thesis by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1166
qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1167
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1168
lemma DERIV_isconst_all:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1169
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1170
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1171
apply (rule linorder_cases [of x y])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1172
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1173
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1174
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1175
lemma DERIV_const_ratio_const:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1176
  fixes f :: "real => real"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1177
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1178
apply (rule linorder_cases [of a b], auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1179
apply (drule_tac [!] f = f in MVT)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1180
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23441
diff changeset
  1181
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1182
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1183
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1184
lemma DERIV_const_ratio_const2:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1185
  fixes f :: "real => real"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1186
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1187
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1188
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1189
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1190
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1191
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1192
by (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1193
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1194
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1195
by (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1196
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1197
text{*Gallileo's "trick": average velocity = av. of end velocities*}
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
lemma DERIV_const_average:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1200
  fixes v :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1201
  assumes neq: "a \<noteq> (b::real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1202
      and der: "\<forall>x. DERIV v x :> k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1203
  shows "v ((a + b)/2) = (v a + v b)/2"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1204
proof (cases rule: linorder_cases [of a b])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1205
  case equal with neq show ?thesis by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1206
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1207
  case less
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1208
  have "(v b - v a) / (b - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1209
    by (rule DERIV_const_ratio_const2 [OF neq der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1210
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1211
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1212
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1213
  ultimately show ?thesis using neq by force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1214
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1215
  case greater
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1216
  have "(v b - v a) / (b - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1217
    by (rule DERIV_const_ratio_const2 [OF neq der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1218
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1219
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1220
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1221
  ultimately show ?thesis using neq by (force simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1222
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1223
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1224
(* A function with positive derivative is increasing. 
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1225
   A simple proof using the MVT, by Jeremy Avigad. And variants.
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1226
*)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1227
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1228
lemma DERIV_pos_imp_increasing:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1229
  fixes a::real and b::real and f::"real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1230
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1231
  shows "f a < f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1232
proof (rule ccontr)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1233
  assume "~ f a < f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1234
  from assms have "EX l z. a < z & z < b & DERIV f z :> l
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1235
      & f b - f a = (b - a) * l"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1236
    by (metis MVT DERIV_isCont differentiableI real_less_def)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1237
  then obtain l z where "a < z" and "z < b" and "DERIV f z :> l"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1238
      and "f b - f a = (b - a) * l"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1239
    by auto
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1240
  
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1241
  from prems have "~(l > 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1242
    by (metis assms(1) linorder_not_le mult_le_0_iff real_le_eq_diff)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1243
  with prems show False
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1244
    by (metis DERIV_unique real_less_def)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1245
qed
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1246
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1247
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1248
lemma DERIV_nonneg_imp_nonincreasing:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1249
  fixes a::real and b::real and f::"real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1250
  assumes "a \<le> b" and
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1251
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1252
  shows "f a \<le> f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1253
proof (rule ccontr, cases "a = b")
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1254
  assume "~ f a \<le> f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1255
  assume "a = b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1256
  with prems show False by auto
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1257
  next assume "~ f a \<le> f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1258
  assume "a ~= b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1259
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1260
      & f b - f a = (b - a) * l"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1261
    apply (intro MVT)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1262
    apply auto
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1263
    apply (metis DERIV_isCont)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1264
    apply (metis differentiableI real_less_def)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1265
    done
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1266
  then obtain l z where "a < z" and "z < b" and "DERIV f z :> l"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1267
      and "f b - f a = (b - a) * l"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1268
    by auto
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1269
  from prems have "~(l >= 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1270
    by (metis diff_self le_eqI le_iff_diff_le_0 real_le_anti_sym real_le_linear
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1271
              split_mult_pos_le)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1272
  with prems show False
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1273
    by (metis DERIV_unique order_less_imp_le)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1274
qed
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1275
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1276
lemma DERIV_neg_imp_decreasing:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1277
  fixes a::real and b::real and f::"real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1278
  assumes "a < b" and
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1279
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1280
  shows "f a > f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1281
proof -
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1282
  have "(%x. -f x) a < (%x. -f x) b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1283
    apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1284
    apply (insert prems, auto)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1285
    apply (metis DERIV_minus neg_0_less_iff_less)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1286
    done
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1287
  thus ?thesis
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1288
    by simp
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1289
qed
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1290
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1291
lemma DERIV_nonpos_imp_nonincreasing:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1292
  fixes a::real and b::real and f::"real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1293
  assumes "a \<le> b" and
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1294
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1295
  shows "f a \<ge> f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1296
proof -
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1297
  have "(%x. -f x) a \<le> (%x. -f x) b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1298
    apply (rule DERIV_nonneg_imp_nonincreasing [of a b "%x. -f x"])
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1299
    apply (insert prems, auto)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1300
    apply (metis DERIV_minus neg_0_le_iff_le)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1301
    done
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1302
  thus ?thesis
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1303
    by simp
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1304
qed
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1305
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
  1306
subsection {* Continuous injective functions *}
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
  1307
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1308
text{*Dull lemma: an continuous injection on an interval must have a
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1309
strict maximum at an end point, not in the middle.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1310
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1311
lemma lemma_isCont_inj:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1312
  fixes f :: "real \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1313
  assumes d: "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1314
      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1315
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1316
  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1317
proof (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1318
  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1319
  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1320
  show False
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1321
  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1322
    case le
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1323
    from d cont all [of "x+d"]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1324
    have flef: "f(x+d) \<le> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1325
     and xlex: "x - d \<le> x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1326
     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1327
       by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1328
    from IVT [OF le flef xlex cont']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1329
    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1330
    moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1331
    hence "g(f x') = g (f(x+d))" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1332
    ultimately show False using d inj [of x'] inj [of "x+d"]
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22984
diff changeset
  1333
      by (simp add: abs_le_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1334
  next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1335
    case ge
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1336
    from d cont all [of "x-d"]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1337
    have flef: "f(x-d) \<le> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1338
     and xlex: "x \<le> x+d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1339
     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1340
       by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1341
    from IVT2 [OF ge flef xlex cont']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1342
    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1343
    moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1344
    hence "g(f x') = g (f(x-d))" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1345
    ultimately show False using d inj [of x'] inj [of "x-d"]
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22984
diff changeset
  1346
      by (simp add: abs_le_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1347
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1348
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1349
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1350
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1351
text{*Similar version for lower bound.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1352
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1353
lemma lemma_isCont_inj2:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1354
  fixes f g :: "real \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1355
  shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1356
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1357
      ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1358
apply (insert lemma_isCont_inj
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1359
          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1360
apply (simp add: isCont_minus linorder_not_le)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1361
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1362
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1363
text{*Show there's an interval surrounding @{term "f(x)"} in
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1364
@{text "f[[x - d, x + d]]"} .*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1365
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1366
lemma isCont_inj_range:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1367
  fixes f :: "real \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1368
  assumes d: "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1369
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1370
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1371
  shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1372
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1373
  have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22984
diff changeset
  1374
    by (auto simp add: abs_le_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1375
  from isCont_Lb_Ub [OF this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1376
  obtain L M
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1377
  where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1378
    and all2 [rule_format]:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1379
           "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1380
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1381
  with d have "L \<le> f x & f x \<le> M" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1382
  moreover have "L \<noteq> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1383
  proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1384
    from lemma_isCont_inj2 [OF d inj cont]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1385
    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1386
    thus ?thesis using all1 [of u] by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1387
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1388
  moreover have "f x \<noteq> M"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1389
  proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1390
    from lemma_isCont_inj [OF d inj cont]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1391
    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1392
    thus ?thesis using all1 [of u] by arith
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
  ultimately have "L < f x & f x < M" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1395
  hence "0 < f x - L" "0 < M - f x" by arith+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1396
  from real_lbound_gt_zero [OF this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1397
  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1398
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1399
  proof (intro exI conjI)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1400
    show "0<e" using e(1) .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1401
    show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1402
    proof (intro strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1403
      fix y::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset