src/HOL/Deriv.thy
author huffman
Tue, 13 Jan 2009 08:19:14 -0800
changeset 29472 a63a2e46cec9
parent 29470 1851088a1f87
child 29667 53103fc8ffa3
permissions -rw-r--r--
declare smult rules [simp]
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(*  Title       : Deriv.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    GMVT by Benjamin Porter, 2005
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*)
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header{* Differentiation *}
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theory Deriv
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imports Lim Polynomial
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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)
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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_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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apply (induct "n")
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apply (auto intro: DERIV_add)
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done
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text{*Alternative definition for differentiability*}
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lemma DERIV_LIM_iff:
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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)"
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by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
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lemma inverse_diff_inverse:
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  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
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   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23441
diff changeset
   144
by (simp add: ring_simps)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   145
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   146
lemma DERIV_inverse_lemma:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   147
  "\<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
   148
   \<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
   149
     = - (inverse a * ((a - b) / h) * inverse b)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   150
by (simp add: inverse_diff_inverse)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   151
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   152
lemma DERIV_inverse':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   153
  assumes der: "DERIV f x :> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   154
  assumes neq: "f x \<noteq> 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   155
  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
   156
    (is "DERIV _ _ :> ?E")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   157
proof (unfold DERIV_iff2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   158
  from der have lim_f: "f -- x --> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   159
    by (rule DERIV_isCont [unfolded isCont_def])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   160
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   161
  from neq have "0 < norm (f x)" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   162
  with LIM_D [OF lim_f] obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   163
    where s: "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   164
    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
   165
                  \<Longrightarrow> norm (f z - f x) < norm (f x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   166
    by fast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   167
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   168
  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
   169
  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
   170
    fix z
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   171
    assume "z \<noteq> x" "norm (z - x) < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   172
    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
   173
    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
   174
    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
   175
          - (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
   176
      using neq by (rule DERIV_inverse_lemma)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   177
  next
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   178
    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
   179
      by (unfold DERIV_iff2)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   180
    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
   181
          -- x --> ?E"
22613
2f119f54d150 remove redundant lemmas
huffman
parents: 21810
diff changeset
   182
      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
   183
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   184
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   185
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   186
lemma DERIV_divide:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   187
  "\<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
   188
   \<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
   189
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
   190
          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
   191
apply (erule subst)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   192
apply (unfold divide_inverse)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   193
apply (erule DERIV_mult')
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   194
apply (erule (1) DERIV_inverse')
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23441
diff changeset
   195
apply (simp add: ring_distribs nonzero_inverse_mult_distrib)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   196
apply (simp add: mult_ac)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   197
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   198
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   199
lemma DERIV_power_Suc:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   200
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   201
  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
   202
  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
   203
proof (induct n)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   204
case 0
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   205
  show ?case by (simp add: power_Suc f)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   206
case (Suc k)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   207
  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
   208
    apply (simp only: of_nat_Suc ring_distribs mult_1_left)
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
   209
    apply (simp only: power_Suc right_distrib mult_ac add_ac)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   210
    done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   211
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   212
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   213
lemma DERIV_power:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   214
  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   215
  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
   216
  shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   217
by (cases "n", simp, simp add: DERIV_power_Suc f)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   218
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   219
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   220
(* ------------------------------------------------------------------------ *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   221
(* Caratheodory formulation of derivative at a point: standard proof        *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   222
(* ------------------------------------------------------------------------ *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   223
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   224
lemma CARAT_DERIV:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   225
     "(DERIV f x :> l) =
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   226
      (\<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
   227
      (is "?lhs = ?rhs")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   228
proof
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   229
  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
   230
  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
   231
  proof (intro exI conjI)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   232
    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
   233
    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
   234
    show "isCont ?g x" using der
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   235
      by (simp add: isCont_iff DERIV_iff diff_minus
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   236
               cong: LIM_equal [rule_format])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   237
    show "?g x = l" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   238
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   239
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   240
  assume "?rhs"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   241
  then obtain g where
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   242
    "(\<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
   243
  thus "(DERIV f x :> l)"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23412
diff changeset
   244
     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
   245
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   246
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   247
lemma DERIV_chain':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   248
  assumes f: "DERIV f x :> D"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   249
  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
   250
  shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   251
proof (unfold DERIV_iff2)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   252
  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
   253
    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
   254
    using CARAT_DERIV [THEN iffD1, OF g] by fast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   255
  from f have "f -- x --> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   256
    by (rule DERIV_isCont [unfolded isCont_def])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   257
  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
   258
    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
   259
  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
   260
          -- x --> d (f x) * D"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   261
    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
   262
  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
   263
    by (simp add: d dfx real_scaleR_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   264
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   265
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   266
(* let's do the standard proof though theorem *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   267
(* LIM_mult2 follows from a NS proof          *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   268
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   269
lemma DERIV_cmult:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   270
      "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
   271
by (drule DERIV_mult' [OF DERIV_const], simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   272
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   273
(* standard version *)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   274
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   275
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   276
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   277
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   278
by (auto dest: DERIV_chain simp add: o_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   279
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   280
(*derivative of linear multiplication*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   281
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 -1*}
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lemma DERIV_inverse:
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  fixes x :: "'a::{real_normed_field,recpower}"
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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 add: power_Suc)
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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,recpower}"
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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 power_Suc nonzero_inverse_mult_distrib)
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text{*Derivative of quotient*}
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lemma DERIV_quotient:
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   305
  fixes x :: "'a::{real_normed_field,recpower}"
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parents: 21404
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   306
  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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   308
by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)
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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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lemma differentiableE [elim?]:
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  assumes "f differentiable x"
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  obtains df where "DERIV f x :> df"
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  using prems unfolding differentiable_def ..
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lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
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by (simp add: differentiable_def)
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lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
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by (force simp add: differentiable_def)
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lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
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  by (rule DERIV_ident [THEN differentiableI])
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lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
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  by (rule DERIV_const [THEN differentiableI])
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lemma differentiable_compose:
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  assumes f: "f differentiable (g x)"
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  assumes g: "g differentiable x"
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  shows "(\<lambda>x. f (g x)) differentiable x"
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proof -
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  from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
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  moreover
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  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
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   343
  ultimately
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  have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
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  thus ?thesis by (rule differentiableI)
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qed
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   347
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   348
lemma differentiable_sum [simp]:
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  assumes "f differentiable x"
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  and "g differentiable x"
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  shows "(\<lambda>x. f x + g x) differentiable x"
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proof -
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   353
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
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   354
  moreover
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   355
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
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   356
  ultimately
6a5f1d8d7344 more proofs about differentiable
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   357
  have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
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   358
  thus ?thesis by (rule differentiableI)
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   359
qed
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   360
6a5f1d8d7344 more proofs about differentiable
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   361
lemma differentiable_minus [simp]:
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   362
  assumes "f differentiable x"
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   363
  shows "(\<lambda>x. - f x) differentiable x"
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parents: 29166
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   364
proof -
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   365
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
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   366
  hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
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   367
  thus ?thesis by (rule differentiableI)
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qed
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   369
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   370
lemma differentiable_diff [simp]:
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  assumes "f differentiable x"
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   372
  assumes "g differentiable x"
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   373
  shows "(\<lambda>x. f x - g x) differentiable x"
29169
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   374
  unfolding diff_minus using prems by simp
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   375
6a5f1d8d7344 more proofs about differentiable
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   376
lemma differentiable_mult [simp]:
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   377
  assumes "f differentiable x"
6a5f1d8d7344 more proofs about differentiable
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diff changeset
   378
  assumes "g differentiable x"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
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   379
  shows "(\<lambda>x. f x * g x) differentiable x"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   380
proof -
29169
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   381
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
21164
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parents:
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   382
  moreover
29169
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parents: 29166
diff changeset
   383
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
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   384
  ultimately
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   385
  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
   386
  thus ?thesis by (rule differentiableI)
21164
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parents:
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   387
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   388
29169
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   389
lemma differentiable_inverse [simp]:
6a5f1d8d7344 more proofs about differentiable
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   390
  assumes "f differentiable x" and "f x \<noteq> 0"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   391
  shows "(\<lambda>x. inverse (f x)) differentiable x"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   392
proof -
29169
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   393
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   394
  hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   395
    using `f x \<noteq> 0` by (rule DERIV_inverse')
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   396
  thus ?thesis by (rule differentiableI)
21164
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parents:
diff changeset
   397
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   398
29169
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diff changeset
   399
lemma differentiable_divide [simp]:
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
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   400
  assumes "f differentiable x"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   401
  assumes "g differentiable x" and "g x \<noteq> 0"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   402
  shows "(\<lambda>x. f x / g x) differentiable x"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   403
  unfolding divide_inverse using prems by simp
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   404
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   405
lemma differentiable_power [simp]:
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   406
  fixes f :: "'a::{recpower,real_normed_field} \<Rightarrow> 'a"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   407
  assumes "f differentiable x"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   408
  shows "(\<lambda>x. f x ^ n) differentiable x"
6a5f1d8d7344 more proofs about differentiable
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parents: 29166
diff changeset
   409
  by (induct n, simp, simp add: power_Suc prems)
6a5f1d8d7344 more proofs about differentiable
huffman
parents: 29166
diff changeset
   410
22984
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parents: 22653
diff changeset
   411
21164
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diff changeset
   412
subsection {* Nested Intervals and Bisection *}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   413
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   414
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   415
     All considerably tidied by lcp.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   416
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   417
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
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parents:
diff changeset
   418
apply (induct "no")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   419
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   420
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   421
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   422
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   423
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   424
         \<forall>n. f(n) \<le> g(n) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
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parents:
diff changeset
   425
      ==> Bseq (f :: nat \<Rightarrow> real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   426
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
   427
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   428
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   429
apply (rule_tac y = "g (Suc na)" in order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   430
apply (induct_tac [2] "na")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   431
apply (auto intro: order_trans)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   432
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   433
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   434
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
   435
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   436
         \<forall>n. f(n) \<le> g(n) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   437
      ==> Bseq (g :: nat \<Rightarrow> real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   438
apply (subst Bseq_minus_iff [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   439
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
   440
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   441
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   442
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   443
lemma f_inc_imp_le_lim:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   444
  fixes f :: "nat \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   445
  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
   446
apply (rule linorder_not_less [THEN iffD1])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   447
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   448
apply (drule real_less_sum_gt_zero)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   449
apply (drule_tac x = "f n + - lim f" in spec, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   450
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
   451
apply (subgoal_tac "lim f \<le> f (no + n) ")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   452
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
   453
apply (auto simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   454
apply (induct_tac "no")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   455
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   456
apply (auto intro: order_trans simp add: diff_minus abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   457
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   458
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   459
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   460
apply (rule LIMSEQ_minus [THEN limI])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   461
apply (simp add: convergent_LIMSEQ_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   462
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   463
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   464
lemma g_dec_imp_lim_le:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   465
  fixes g :: "nat \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   466
  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
   467
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   468
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
   469
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
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 lemma_nest: "[| \<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
      ==> \<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
   476
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   477
apply (subgoal_tac "monoseq f & monoseq g")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   478
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   479
apply (subgoal_tac "Bseq f & Bseq g")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   480
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
   481
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   482
apply (rule_tac x = "lim f" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   483
apply (rule_tac x = "lim g" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   484
apply (auto intro: LIMSEQ_le)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   485
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
   486
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   487
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   488
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
   489
         \<forall>n. g(Suc n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   490
         \<forall>n. f(n) \<le> g(n);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   491
         (%n. f(n) - g(n)) ----> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   492
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   493
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   494
apply (drule lemma_nest, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   495
apply (subgoal_tac "l = m")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   496
apply (drule_tac [2] X = f in LIMSEQ_diff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   497
apply (auto intro: LIMSEQ_unique)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   498
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   499
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   500
text{*The universal quantifiers below are required for the declaration
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   501
  of @{text Bolzano_nest_unique} below.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   502
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   503
lemma Bolzano_bisect_le:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   504
 "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
   505
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   506
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   507
apply (auto simp add: Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   508
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   509
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   510
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   511
   \<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
   512
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   513
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   514
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   515
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   516
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   517
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   518
   \<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
   519
apply (rule allI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   520
apply (induct_tac "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   521
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   522
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   523
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   524
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
   525
apply (auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   526
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   527
apply (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   528
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   529
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   530
lemma Bolzano_bisect_diff:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   531
     "a \<le> b ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   532
      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
   533
      (b-a) / (2 ^ n)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   534
apply (induct "n")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   535
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
   536
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   537
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   538
lemmas Bolzano_nest_unique =
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   539
    lemma_nest_unique
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   540
    [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
   541
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 not_P_Bolzano_bisect:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   544
  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
   545
      and notP: "~ P(a,b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   546
      and le:   "a \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   547
  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
   548
proof (induct n)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   549
  case 0 show ?case using notP by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   550
 next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   551
  case (Suc n)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   552
  thus ?case
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   553
 by (auto simp del: surjective_pairing [symmetric]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   554
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   555
     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
   556
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   557
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   558
(*Now we re-package P_prem as a formula*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   559
lemma not_P_Bolzano_bisect':
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   560
     "[| \<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
   561
         ~ P(a,b);  a \<le> b |] ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   562
      \<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
   563
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   564
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   565
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   566
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   567
lemma lemma_BOLZANO:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   568
     "[| \<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
   569
         \<forall>x. \<exists>d::real. 0 < d &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   570
                (\<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
   571
         a \<le> b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   572
      ==> P(a,b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   573
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   574
apply (rule LIMSEQ_minus_cancel)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   575
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
   576
apply (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   577
apply (drule not_P_Bolzano_bisect', assumption+)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   578
apply (rename_tac "l")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   579
apply (drule_tac x = l in spec, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   580
apply (simp add: LIMSEQ_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   581
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
   582
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
   583
apply (drule real_less_half_sum, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   584
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
   585
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
   586
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   587
apply (simp_all (no_asm_simp))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   588
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
   589
apply (simp (no_asm_simp) add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   590
apply (rule real_sum_of_halves [THEN subst])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   591
apply (rule add_strict_mono)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   592
apply (simp_all add: diff_minus [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   593
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   594
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   595
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   596
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
   597
       (\<forall>x. \<exists>d::real. 0 < d &
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   598
                (\<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
   599
      --> (\<forall>a b. a \<le> b --> P(a,b))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   600
apply clarify
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   601
apply (blast intro: lemma_BOLZANO)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   602
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   603
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
subsection {* Intermediate Value Theorem *}
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
text {*Prove Contrapositive by Bisection*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   608
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   609
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
   610
         a \<le> b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   611
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   612
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   613
apply (rule contrapos_pp, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   614
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
   615
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   616
apply simp_all
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   617
apply (simp add: isCont_iff LIM_def)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   618
apply (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   619
apply (subgoal_tac "a \<le> x & x \<le> b")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   620
 prefer 2
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   621
 apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   622
 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
   623
apply (drule_tac x = x in spec)+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   624
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   625
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
   626
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   627
apply simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   628
apply (drule_tac x = s in spec, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   629
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
   630
apply (drule_tac x = "ba-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   631
apply (simp_all add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   632
apply (drule_tac x = "aa-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   633
apply (case_tac "x \<le> aa", simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   634
done
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 IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   637
         a \<le> b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   638
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   639
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   640
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
   641
apply (drule IVT [where f = "%x. - f x"], assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   642
apply (auto intro: isCont_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   643
done
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
(*HOL style here: object-level formulations*)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   646
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
   647
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   648
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   649
apply (blast intro: IVT)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   650
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   651
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   652
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
   653
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   654
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   655
apply (blast intro: IVT2)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   656
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   657
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   658
text{*By bisection, function continuous on closed interval is bounded above*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   659
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   660
lemma isCont_bounded:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   661
     "[| 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
   662
      ==> \<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
   663
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
   664
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   665
apply simp_all
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   666
apply (rename_tac x xa ya M Ma)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   667
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   668
apply (rule_tac x = Ma in exI, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   669
apply (cut_tac x = xb and y = xa in linorder_linear, force)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   670
apply (rule_tac x = M in exI, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   671
apply (cut_tac x = xb and y = xa in linorder_linear, force)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   672
apply (case_tac "a \<le> x & x \<le> b")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   673
apply (rule_tac [2] x = 1 in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   674
prefer 2 apply force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   675
apply (simp add: LIM_def isCont_iff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   676
apply (drule_tac x = x in spec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   677
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
   678
apply (drule_tac x = 1 in spec, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   679
apply (rule_tac x = s in exI, clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   680
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
   681
apply (drule_tac x = "xa-x" in spec)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   682
apply (auto simp add: abs_ge_self)
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
text{*Refine the above to existence of least upper bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   686
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   687
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
   688
      (\<exists>t. isLub UNIV S t)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   689
by (blast intro: reals_complete)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   690
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   691
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
   692
         ==> \<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
   693
                   (\<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
   694
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
   695
        in lemma_reals_complete)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   696
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   697
apply (drule isCont_bounded, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   698
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
   699
apply (rule exI, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   700
apply (auto dest!: spec simp add: linorder_not_less)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   701
done
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
text{*Now show that it attains its upper bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   704
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   705
lemma isCont_eq_Ub:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   706
  assumes le: "a \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   707
      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
   708
  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
   709
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   710
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   711
  from isCont_has_Ub [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   712
  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
   713
             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
   714
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   715
  proof (intro exI, intro conjI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   716
    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
   717
    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
   718
    proof (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   719
      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
   720
      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
   721
        by (fastsimp simp add: linorder_not_le [symmetric])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   722
      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
   723
        by (auto simp add: isCont_inverse isCont_diff con)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   724
      from isCont_bounded [OF le this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   725
      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
   726
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   727
        by (simp add: M3 compare_rls)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   728
      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
   729
        by (auto intro: order_le_less_trans [of _ k])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   730
      with Minv
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   731
      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
   732
        by (intro strip less_imp_inverse_less, simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   733
      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
   734
        by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   735
      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
   736
        by (simp, arith)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   737
      from M2 [OF this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   738
      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
   739
      thus False using invlt [of x] by force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   740
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   741
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   742
qed
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
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   745
text{*Same theorem for lower bound*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   746
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   747
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
   748
         ==> \<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
   749
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   750
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
   751
prefer 2 apply (blast intro: isCont_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   752
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   753
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   754
apply auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   755
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   756
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   757
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   758
text{*Another version.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   759
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   760
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
   761
      ==> \<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
   762
          (\<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
   763
apply (frule isCont_eq_Lb)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   764
apply (frule_tac [2] isCont_eq_Ub)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   765
apply (assumption+, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   766
apply (rule_tac x = "f x" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   767
apply (rule_tac x = "f xa" in exI, simp, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   768
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
   769
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
   770
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
   771
apply (rule_tac [2] x = xb in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   772
apply (rule_tac [4] x = xb in exI, simp_all)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   773
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   774
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   775
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   776
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
   777
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   778
lemma DERIV_left_inc:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   779
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   780
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   781
      and l:   "0 < l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   782
  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
   783
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   784
  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
   785
  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
   786
    by (simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   787
  then obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   788
        where s:   "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   789
          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
   790
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   791
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   792
  proof (intro exI conjI strip)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   793
    show "0<s" using s .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   794
    fix h::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   795
    assume "0 < h" "h < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   796
    with all [of h] show "f x < f (x+h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   797
    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
   798
    split add: split_if_asm)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   799
      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
   800
      with l
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   801
      have "0 < (f (x+h) - f x) / h" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   802
      thus "f x < f (x+h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   803
  by (simp add: pos_less_divide_eq h)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   804
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   805
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   806
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   807
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   808
lemma DERIV_left_dec:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   809
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   810
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   811
      and l:   "l < 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   812
  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
   813
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   814
  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
   815
  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
   816
    by (simp add: diff_minus)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   817
  then obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   818
        where s:   "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   819
          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
   820
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   821
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   822
  proof (intro exI conjI strip)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   823
    show "0<s" using s .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   824
    fix h::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   825
    assume "0 < h" "h < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   826
    with all [of "-h"] show "f x < f (x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   827
    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
   828
    split add: split_if_asm)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   829
      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
   830
      with l
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   831
      have "0 < (f (x-h) - f x) / h" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   832
      thus "f x < f (x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   833
  by (simp add: pos_less_divide_eq h)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   834
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   835
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   836
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   837
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   838
lemma DERIV_local_max:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   839
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   840
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   841
      and d:   "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   842
      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
   843
  shows "l = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   844
proof (cases rule: linorder_cases [of l 0])
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   845
  case equal thus ?thesis .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   846
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   847
  case less
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   848
  from DERIV_left_dec [OF der less]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   849
  obtain d' where d': "0 < d'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   850
             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
   851
  from real_lbound_gt_zero [OF d d']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   852
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   853
  with lt le [THEN spec [where x="x-e"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   854
  show ?thesis by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   855
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   856
  case greater
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   857
  from DERIV_left_inc [OF der greater]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   858
  obtain d' where d': "0 < d'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   859
             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
   860
  from real_lbound_gt_zero [OF d d']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   861
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   862
  with lt le [THEN spec [where x="x+e"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   863
  show ?thesis by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   864
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   865
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   866
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   867
text{*Similar theorem for a local minimum*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   868
lemma DERIV_local_min:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   869
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   870
  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
   871
by (drule DERIV_minus [THEN DERIV_local_max], auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   872
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   873
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   874
text{*In particular, if a function is locally flat*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   875
lemma DERIV_local_const:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   876
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   877
  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
   878
by (auto dest!: DERIV_local_max)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   879
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   880
text{*Lemma about introducing open ball in open interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   881
lemma lemma_interval_lt:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   882
     "[| a < x;  x < b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   883
      ==> \<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
   884
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22984
diff changeset
   885
apply (simp add: abs_less_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   886
apply (insert linorder_linear [of "x-a" "b-x"], safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   887
apply (rule_tac x = "x-a" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   888
apply (rule_tac [2] x = "b-x" in exI, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   889
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   890
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   891
lemma lemma_interval: "[| a < x;  x < b |] ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   892
        \<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
   893
apply (drule lemma_interval_lt, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   894
apply (auto intro!: exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   895
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   896
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   897
text{*Rolle's Theorem.
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   898
   If @{term f} is defined and continuous on the closed interval
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   899
   @{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
   900
   and @{term "f(a) = f(b)"},
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   901
   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
   902
theorem Rolle:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   903
  assumes lt: "a < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   904
      and eq: "f(a) = f(b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   905
      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
   906
      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
   907
  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
   908
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   909
  have le: "a \<le> b" using lt by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   910
  from isCont_eq_Ub [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   911
  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
   912
             and alex: "a \<le> x" and xleb: "x \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   913
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   914
  from isCont_eq_Lb [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   915
  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
   916
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   917
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   918
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   919
  proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   920
    assume axb: "a < x & x < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   921
        --{*@{term f} attains its maximum within the interval*}
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
   922
    hence ax: "a<x" and xb: "x<b" by arith + 
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   923
    from lemma_interval [OF ax xb]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   924
    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
   925
      by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   926
    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
   927
      by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   928
    from differentiableD [OF dif [OF axb]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   929
    obtain l where der: "DERIV f x :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   930
    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
   931
        --{*the derivative at a local maximum is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   932
    thus ?thesis using ax xb der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   933
  next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   934
    assume notaxb: "~ (a < x & x < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   935
    hence xeqab: "x=a | x=b" using alex xleb by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   936
    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
   937
    show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   938
    proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   939
      assume ax'b: "a < x' & x' < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   940
        --{*@{term f} attains its minimum within the interval*}
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
   941
      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
   942
      from lemma_interval [OF ax' x'b]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   943
      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
   944
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   945
      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
   946
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   947
      from differentiableD [OF dif [OF ax'b]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   948
      obtain l where der: "DERIV f x' :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   949
      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
   950
        --{*the derivative at a local minimum is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   951
      thus ?thesis using ax' x'b der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   952
    next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   953
      assume notax'b: "~ (a < x' & x' < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   954
        --{*@{term f} is constant througout the interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   955
      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
   956
      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
   957
      from dense [OF lt]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   958
      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
   959
      from lemma_interval [OF ar rb]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   960
      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
   961
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   962
      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
   963
      proof (clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   964
        fix z::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   965
        assume az: "a \<le> z" and zb: "z \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   966
        show "f z = f b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   967
        proof (rule order_antisym)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   968
          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
   969
          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
   970
        qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   971
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   972
      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
   973
      proof (intro strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   974
        fix y::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   975
        assume lt: "\<bar>r-y\<bar> < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   976
        hence "f y = f b" by (simp add: eq_fb bound)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   977
        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
   978
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   979
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   980
      obtain l where der: "DERIV f r :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   981
      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
   982
        --{*the derivative of a constant function is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   983
      thus ?thesis using ar rb der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   984
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   985
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   986
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   987
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   988
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   989
subsection{*Mean Value Theorem*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   990
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   991
lemma lemma_MVT:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   992
     "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
   993
proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   994
  assume "a=b" thus ?thesis by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   995
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   996
  assume "a\<noteq>b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   997
  hence ba: "b-a \<noteq> 0" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   998
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   999
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1000
        simp add: right_diff_distrib,
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1001
        simp add: left_diff_distrib)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1002
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1003
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1004
theorem MVT:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1005
  assumes lt:  "a < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1006
      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
  1007
      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
  1008
  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
  1009
                   (f(b) - f(a) = (b-a) * l)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1010
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1011
  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
  1012
  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
  1013
    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
  1014
  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
  1015
  proof (clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1016
    fix x::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1017
    assume ax: "a < x" and xb: "x < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1018
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1019
    obtain l where der: "DERIV f x :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1020
    show "?F differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1021
      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
  1022
          blast intro: DERIV_diff DERIV_cmult_Id der)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1023
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1024
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1025
  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
  1026
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1027
  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
  1028
    by (rule DERIV_cmult_Id)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1029
  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
  1030
                   :> 0 + (f b - f a) / (b - a)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1031
    by (rule DERIV_add [OF der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1032
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1033
  proof (intro exI conjI)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1034
    show "a < z" using az .
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1035
    show "z < b" using zb .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1036
    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
  1037
    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
  1038
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1039
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1040
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1041
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1042
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
  1043
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1044
lemma DERIV_isconst_end:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1045
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1046
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1047
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1048
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1049
        ==> f b = f a"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1050
apply (drule MVT, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1051
apply (blast intro: differentiableI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1052
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1053
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1054
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1055
lemma DERIV_isconst1:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1056
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1057
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1058
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1059
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1060
        ==> \<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
  1061
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1062
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
  1063
apply (drule_tac b = x in DERIV_isconst_end, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1064
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1065
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1066
lemma DERIV_isconst2:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1067
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1068
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1069
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1070
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1071
         a \<le> x; x \<le> b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1072
        ==> f x = f a"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1073
apply (blast dest: DERIV_isconst1)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1074
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1075
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1076
lemma DERIV_isconst_all:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1077
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1078
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1079
apply (rule linorder_cases [of x y])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1080
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1081
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1082
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1083
lemma DERIV_const_ratio_const:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1084
  fixes f :: "real => real"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1085
  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
  1086
apply (rule linorder_cases [of a b], auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1087
apply (drule_tac [!] f = f in MVT)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1088
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
  1089
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
  1090
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1091
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1092
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
  1093
  fixes f :: "real => real"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1094
  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
  1095
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
  1096
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1097
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1098
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1099
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
  1100
by (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1101
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1102
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
  1103
by (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1104
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1105
text{*Gallileo's "trick": average velocity = av. of end velocities*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1106
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1107
lemma DERIV_const_average:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1108
  fixes v :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1109
  assumes neq: "a \<noteq> (b::real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1110
      and der: "\<forall>x. DERIV v x :> k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1111
  shows "v ((a + b)/2) = (v a + v b)/2"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1112
proof (cases rule: linorder_cases [of a b])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1113
  case equal with neq show ?thesis by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1114
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1115
  case less
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1116
  have "(v b - v a) / (b - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1117
    by (rule DERIV_const_ratio_const2 [OF neq der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1118
  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
  1119
  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
  1120
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1121
  ultimately show ?thesis using neq by force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1122
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1123
  case greater
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1124
  have "(v b - v a) / (b - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1125
    by (rule DERIV_const_ratio_const2 [OF neq der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1126
  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
  1127
  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
  1128
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1129
  ultimately show ?thesis using neq by (force simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1130
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1131
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1132
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1133
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
  1134
strict maximum at an end point, not in the middle.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1135
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1136
lemma lemma_isCont_inj:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1137
  fixes f :: "real \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1138
  assumes d: "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1139
      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
  1140
      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
  1141
  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
  1142
proof (rule ccontr)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1143
  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
  1144
  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
  1145
  show False
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1146
  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
  1147
    case le
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1148
    from d cont all [of "x+d"]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1149
    have flef: "f(x+d) \<le> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1150
     and xlex: "x - d \<le> x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1151
     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
  1152
       by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1153
    from IVT [OF le flef xlex cont']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1154
    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
  1155
    moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1156
    hence "g(f x') = g (f(x+d))" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1157
    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
  1158
      by (simp add: abs_le_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1159
  next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1160
    case ge
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1161
    from d cont all [of "x-d"]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1162
    have flef: "f(x-d) \<le> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1163
     and xlex: "x \<le> x+d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1164
     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
  1165
       by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1166
    from IVT2 [OF ge flef xlex cont']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1167
    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
  1168
    moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1169
    hence "g(f x') = g (f(x-d))" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1170
    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
  1171
      by (simp add: abs_le_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1172
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1173
qed
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
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1176
text{*Similar version for lower bound.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1177
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1178
lemma lemma_isCont_inj2:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1179
  fixes f g :: "real \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1180
  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
  1181
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1182
      ==> \<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
  1183
apply (insert lemma_isCont_inj
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1184
          [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
  1185
apply (simp add: isCont_minus linorder_not_le)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1186
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1187
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1188
text{*Show there's an interval surrounding @{term "f(x)"} in
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1189
@{text "f[[x - d, x + d]]"} .*}
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 isCont_inj_range:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1192
  fixes f :: "real \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1193
  assumes d: "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1194
      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
  1195
      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
  1196
  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
  1197
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1198
  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
  1199
    by (auto simp add: abs_le_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1200
  from isCont_Lb_Ub [OF this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1201
  obtain L M
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1202
  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
  1203
    and all2 [rule_format]:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1204
           "\<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
  1205
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1206
  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
  1207
  moreover have "L \<noteq> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1208
  proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1209
    from lemma_isCont_inj2 [OF d inj cont]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1210
    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
  1211
    thus ?thesis using all1 [of u] by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1212
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1213
  moreover have "f x \<noteq> M"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1214
  proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1215
    from lemma_isCont_inj [OF d inj cont]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1216
    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
  1217
    thus ?thesis using all1 [of u] by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1218
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1219
  ultimately have "L < f x & f x < M" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1220
  hence "0 < f x - L" "0 < M - f x" by arith+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1221
  from real_lbound_gt_zero [OF this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1222
  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
  1223
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1224
  proof (intro exI conjI)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1225
    show "0<e" using e(1) .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1226
    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
  1227
    proof (intro strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1228
      fix y::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1229
      assume "\<bar>y - f x\<bar> \<le> e"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1230
      with e have "L \<le> y \<and> y \<le> M" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1231
      from all2 [OF this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1232
      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
  1233
      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y" 
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22984
diff changeset
  1234
        by (force simp add: abs_le_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1235
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1236
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1237
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1238
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1239
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1240
text{*Continuity of inverse function*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1241
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1242
lemma isCont_inverse_function:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1243
  fixes f g :: "real \<Rightarrow> real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1244
  assumes d: "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1245
      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
  1246
      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
  1247
  shows "isCont g (f x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1248
proof (simp add: isCont_iff LIM_eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1249
  show "\<forall>r. 0 < r \<longrightarrow>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1250
         (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1251
  proof (intro strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1252
    fix r::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1253
    assume r: "0<r"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1254
    from real_lbound_gt_zero [OF r d]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1255
    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1256
    with inj cont
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1257
    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1258
                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1259
    from isCont_inj_range [OF e this]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1260
    obtain e' where e': "0 < e'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1261
        and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1262
          by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1263
    show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1264
    proof (intro exI conjI)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1265
      show "0<e'" using e' .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1266
      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1267
      proof (intro strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1268
        fix z::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1269
        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1270
        with e e_lt e_simps all [rule_format, of "f x + z"]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1271
        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1272
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1273
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1274
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1275
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1276
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1277
text {* Derivative of inverse function *}
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1278
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1279
lemma DERIV_inverse_function:
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1280
  fixes f g :: "real \<Rightarrow> real"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1281
  assumes der: "DERIV f (g x) :> D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1282
  assumes neq: "D \<noteq> 0"
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1283
  assumes a: "a < x" and b: "x < b"
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1284
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1285
  assumes cont: "isCont g x"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1286
  shows "DERIV g x :> inverse D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1287
unfolding DERIV_iff2
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1288
proof (rule LIM_equal2)
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1289
  show "0 < min (x - a) (b - x)"
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
  1290
    using a b by arith 
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1291
next
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1292
  fix y
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1293
  assume "norm (y - x) < min (x - a) (b - x)"
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
  1294
  hence "a < y" and "y < b" 
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1295
    by (simp_all add: abs_less_iff)
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1296
  thus "(g y - g x) / (y - x) =
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1297
        inverse ((f (g y) - x) / (g y - g x))"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1298
    by (simp add: inj)
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1299
next
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1300
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1301
    by (rule der [unfolded DERIV_iff2])
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1302
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1303
    using inj a b by simp
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1304
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1305
  proof (safe intro!: exI)
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1306
    show "0 < min (x - a) (b - x)"
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1307
      using a b by simp
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1308
  next
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1309
    fix y
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1310
    assume "norm (y - x) < min (x - a) (b - x)"
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1311
    hence y: "a < y" "y < b"
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1312
      by (simp_all add: abs_less_iff)
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1313
    assume "g y = g x"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1314
    hence "f (g y) = f (g x)" by simp
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1315
    hence "y = x" using inj y a b by simp
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1316
    also assume "y \<noteq> x"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1317
    finally show False by simp
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1318
  qed
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1319
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1320
    using cont 1 2 by (rule isCont_LIM_compose2)
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1321
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1322
        -- x --> inverse D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1323
    using neq by (rule LIM_inverse)
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1324
qed
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1325
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1326
theorem GMVT:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1327
  fixes a b :: real
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1328
  assumes alb: "a < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1329
  and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1330
  and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1331
  and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1332
  and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1333
  shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1334
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1335
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1336
  from prems have "a < b" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1337
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1338
  proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1339
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1340
    with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1341
      by (auto intro: isCont_mult)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1342
    moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1343
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1344
    with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1345
      by (auto intro: isCont_mult)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1346
    ultimately show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1347
      by (fastsimp intro: isCont_diff)
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
  moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1350
  have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1351
  proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1352
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1353
    with gd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (f b - f a) * g x) differentiable x" by (simp add: differentiable_mult)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1354
    moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1355
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1356
    with fd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (g b - g a) * f x) differentiable x" by (simp add: differentiable_mult)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1357
    ultimately show ?thesis by (simp add: differentiable_diff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1358
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1359
  ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1360
  then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1361
  then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
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
  from cdef have cint: "a < c \<and> c < b" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1364
  with gd have "g differentiable c" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1365
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1366
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1367
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1368
  from cdef have "a < c \<and> c < b" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1369
  with fd have "f differentiable c" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1370
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1371
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1372
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1373
  from cdef have "DERIV ?h c :> l" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1374
  moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1375
  {
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1376
    have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1377
      apply (insert DERIV_const [where k="f b - f a"])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1378
      apply (drule meta_spec [of _ c])
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1379
      apply (drule DERIV_mult [OF _ g'cdef])
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1380
      by simp
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1381
    moreover have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1382
      apply (insert DERIV_const [where k="g b - g a"])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1383
      apply (drule meta_spec [of _ c])
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1384
      apply (drule DERIV_mult [OF _ f'cdef])
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1385
      by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1386
    ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1387
      by (simp add: DERIV_diff)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1388
  }
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1389
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1390
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1391
  {
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1392
    from cdef have "?h b - ?h a = (b - a) * l" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1393
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1394
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1395
  }
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1396
  moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1397
  {
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1398
    have "?h b - ?h a =
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1399
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1400
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22984
diff changeset
  1401
      by (simp add: mult_ac add_ac right_diff_distrib)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1402
    hence "?h b - ?h a = 0" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1403
  }
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1404
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1405
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1406
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1407
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1408
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1409
  with g'cdef f'cdef cint show ?thesis by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1410
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1411
23255
631bd424fd72 lemma lemma_DERIV_subst moved to Deriv.thy
chaieb
parents: 23069
diff changeset
  1412
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
631bd424fd72 lemma lemma_DERIV_subst moved to Deriv.thy
chaieb
parents: 23069
diff changeset
  1413
by auto
631bd424fd72 lemma lemma_DERIV_subst moved to Deriv.thy
chaieb
parents: 23069
diff changeset
  1414
29470
1851088a1f87 convert Deriv.thy to use new Polynomial library (incomplete)
huffman
parents: 29169
diff changeset
  1415
26120
2dd43c63c100 Includes the derivates of polynomials -- reals specific content of Poly
chaieb
parents: 23477
diff changeset
  1416
subsection {* Derivatives of univariate polynomials *}
2dd43c63c100 Includes the derivates of polynomials -- reals specific content of Poly
chaieb
parents: