src/HOL/Deriv.thy
author hoelzl
Mon Mar 17 19:12:52 2014 +0100 (2014-03-17)
changeset 56181 2aa0b19e74f3
parent 55970 6d123f0ae358
child 56182 528fae0816ea
permissions -rw-r--r--
unify syntax for has_derivative and differentiable
huffman@21164
     1
(*  Title       : Deriv.thy
huffman@21164
     2
    Author      : Jacques D. Fleuriot
huffman@21164
     3
    Copyright   : 1998  University of Cambridge
hoelzl@51642
     4
    Author      : Brian Huffman
huffman@21164
     5
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
huffman@21164
     6
    GMVT by Benjamin Porter, 2005
huffman@21164
     7
*)
huffman@21164
     8
huffman@21164
     9
header{* Differentiation *}
huffman@21164
    10
huffman@21164
    11
theory Deriv
hoelzl@51526
    12
imports Limits
huffman@21164
    13
begin
huffman@21164
    14
hoelzl@56181
    15
subsection {* Frechet derivative *}
hoelzl@56181
    16
hoelzl@51642
    17
definition
hoelzl@51642
    18
  has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow>  bool"
hoelzl@51642
    19
  (infixl "(has'_derivative)" 12)
hoelzl@51642
    20
where
hoelzl@51642
    21
  "(f has_derivative f') F \<longleftrightarrow>
hoelzl@51642
    22
    (bounded_linear f' \<and>
hoelzl@51642
    23
     ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) ---> 0) F)"
hoelzl@51642
    24
hoelzl@56181
    25
text {*
hoelzl@56181
    26
  Usually the filter @{term F} is @{term "at x within s"}.  @{term "(f has_derivative D)
hoelzl@56181
    27
  (at x within s)"} means: @{term D} is the derivative of function @{term f} at point @{term x}
hoelzl@56181
    28
  within the set @{term s}. Where @{term s} is used to express left or right sided derivatives. In
hoelzl@56181
    29
  most cases @{term s} is either a variable or @{term UNIV}.
hoelzl@56181
    30
*}
hoelzl@56181
    31
hoelzl@56181
    32
text {*
hoelzl@56181
    33
  The following syntax is only used as a legacy syntax.
hoelzl@56181
    34
*}
hoelzl@56181
    35
abbreviation (input)
hoelzl@56181
    36
  FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
hoelzl@56181
    37
  ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
hoelzl@56181
    38
where
hoelzl@56181
    39
  "FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"
hoelzl@56181
    40
hoelzl@56181
    41
hoelzl@56181
    42
lemma has_derivative_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
hoelzl@51642
    43
  by simp
hoelzl@51642
    44
hoelzl@51642
    45
ML {*
hoelzl@51642
    46
hoelzl@56181
    47
structure has_derivative_Intros = Named_Thms
hoelzl@51642
    48
(
hoelzl@56181
    49
  val name = @{binding has_derivative_intros}
hoelzl@51642
    50
  val description = "introduction rules for FDERIV"
hoelzl@51642
    51
)
hoelzl@51642
    52
hoelzl@51642
    53
*}
hoelzl@51642
    54
hoelzl@51642
    55
setup {*
hoelzl@56181
    56
  has_derivative_Intros.setup #>
hoelzl@56181
    57
  Global_Theory.add_thms_dynamic (@{binding has_derivative_eq_intros},
hoelzl@56181
    58
    map_filter (try (fn thm => @{thm has_derivative_eq_rhs} OF [thm])) o has_derivative_Intros.get o Context.proof_of);
hoelzl@51642
    59
*}
hoelzl@51642
    60
hoelzl@56181
    61
lemma has_derivative_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
hoelzl@51642
    62
  by (simp add: has_derivative_def)
hoelzl@51642
    63
hoelzl@56181
    64
lemma has_derivative_ident[has_derivative_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
hoelzl@51642
    65
  by (simp add: has_derivative_def tendsto_const)
hoelzl@51642
    66
hoelzl@56181
    67
lemma has_derivative_const[has_derivative_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"
hoelzl@51642
    68
  by (simp add: has_derivative_def tendsto_const)
hoelzl@51642
    69
hoelzl@51642
    70
lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
hoelzl@51642
    71
hoelzl@56181
    72
lemma (in bounded_linear) has_derivative:
hoelzl@51642
    73
  "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
hoelzl@51642
    74
  using assms unfolding has_derivative_def
hoelzl@51642
    75
  apply safe
hoelzl@51642
    76
  apply (erule bounded_linear_compose [OF local.bounded_linear])
hoelzl@51642
    77
  apply (drule local.tendsto)
hoelzl@51642
    78
  apply (simp add: local.scaleR local.diff local.add local.zero)
hoelzl@51642
    79
  done
hoelzl@51642
    80
hoelzl@56181
    81
lemmas has_derivative_scaleR_right [has_derivative_intros] =
hoelzl@56181
    82
  bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
hoelzl@51642
    83
hoelzl@56181
    84
lemmas has_derivative_scaleR_left [has_derivative_intros] =
hoelzl@56181
    85
  bounded_linear.has_derivative [OF bounded_linear_scaleR_left]
hoelzl@51642
    86
hoelzl@56181
    87
lemmas has_derivative_mult_right [has_derivative_intros] =
hoelzl@56181
    88
  bounded_linear.has_derivative [OF bounded_linear_mult_right]
hoelzl@51642
    89
hoelzl@56181
    90
lemmas has_derivative_mult_left [has_derivative_intros] =
hoelzl@56181
    91
  bounded_linear.has_derivative [OF bounded_linear_mult_left]
hoelzl@51642
    92
hoelzl@56181
    93
lemma has_derivative_add[simp, has_derivative_intros]:
hoelzl@51642
    94
  assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F"
hoelzl@51642
    95
  shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
hoelzl@51642
    96
  unfolding has_derivative_def
hoelzl@51642
    97
proof safe
hoelzl@51642
    98
  let ?x = "Lim F (\<lambda>x. x)"
hoelzl@51642
    99
  let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
hoelzl@51642
   100
  have "((\<lambda>x. ?D f f' x + ?D g g' x) ---> (0 + 0)) F"
hoelzl@51642
   101
    using f g by (intro tendsto_add) (auto simp: has_derivative_def)
hoelzl@51642
   102
  then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"
hoelzl@51642
   103
    by (simp add: field_simps scaleR_add_right scaleR_diff_right)
hoelzl@56181
   104
qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)
hoelzl@51642
   105
hoelzl@56181
   106
lemma has_derivative_setsum[simp, has_derivative_intros]:
hoelzl@51642
   107
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F"
hoelzl@51642
   108
  shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
hoelzl@51642
   109
proof cases
hoelzl@51642
   110
  assume "finite I" from this f show ?thesis
hoelzl@51642
   111
    by induct (simp_all add: f)
hoelzl@51642
   112
qed simp
hoelzl@51642
   113
hoelzl@56181
   114
lemma has_derivative_minus[simp, has_derivative_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
hoelzl@56181
   115
  using has_derivative_scaleR_right[of f f' F "-1"] by simp
hoelzl@51642
   116
hoelzl@56181
   117
lemma has_derivative_diff[simp, has_derivative_intros]:
hoelzl@56181
   118
  "(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
hoelzl@56181
   119
  by (simp only: diff_conv_add_uminus has_derivative_add has_derivative_minus)
hoelzl@51642
   120
hoelzl@56181
   121
lemma has_derivative_at_within:
hoelzl@56181
   122
  "(f has_derivative f') (at x within s) \<longleftrightarrow>
hoelzl@51642
   123
    (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s))"
hoelzl@51642
   124
  by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)
hoelzl@51642
   125
hoelzl@56181
   126
lemma has_derivative_iff_norm:
hoelzl@56181
   127
  "(f has_derivative f') (at x within s) \<longleftrightarrow>
hoelzl@51642
   128
    (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ---> 0) (at x within s))"
hoelzl@51642
   129
  using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
hoelzl@56181
   130
  by (simp add: has_derivative_at_within divide_inverse ac_simps)
hoelzl@51642
   131
hoelzl@56181
   132
lemma has_derivative_at:
hoelzl@56181
   133
  "(f has_derivative D) (at x) \<longleftrightarrow> (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0)"
hoelzl@56181
   134
  unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp
hoelzl@51642
   135
hoelzl@56181
   136
lemma field_has_derivative_at:
hoelzl@51642
   137
  fixes x :: "'a::real_normed_field"
hoelzl@56181
   138
  shows "(f has_derivative op * D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@56181
   139
  apply (unfold has_derivative_at)
hoelzl@56181
   140
  apply (simp add: bounded_linear_mult_right)
hoelzl@51642
   141
  apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
hoelzl@51642
   142
  apply (subst diff_divide_distrib)
hoelzl@51642
   143
  apply (subst times_divide_eq_left [symmetric])
hoelzl@51642
   144
  apply (simp cong: LIM_cong)
hoelzl@51642
   145
  apply (simp add: tendsto_norm_zero_iff LIM_zero_iff)
hoelzl@51642
   146
  done
hoelzl@51642
   147
hoelzl@56181
   148
lemma has_derivativeI:
hoelzl@51642
   149
  "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s) \<Longrightarrow>
hoelzl@56181
   150
  (f has_derivative f') (at x within s)"
hoelzl@56181
   151
  by (simp add: has_derivative_at_within)
hoelzl@51642
   152
hoelzl@56181
   153
lemma has_derivativeI_sandwich:
hoelzl@51642
   154
  assumes e: "0 < e" and bounded: "bounded_linear f'"
hoelzl@51642
   155
    and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow> norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
hoelzl@51642
   156
    and "(H ---> 0) (at x within s)"
hoelzl@56181
   157
  shows "(f has_derivative f') (at x within s)"
hoelzl@56181
   158
  unfolding has_derivative_iff_norm
hoelzl@51642
   159
proof safe
hoelzl@51642
   160
  show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) ---> 0) (at x within s)"
hoelzl@51642
   161
  proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
hoelzl@51642
   162
    show "(H ---> 0) (at x within s)" by fact
hoelzl@51642
   163
    show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
hoelzl@51642
   164
      unfolding eventually_at using e sandwich by auto
hoelzl@51642
   165
  qed (auto simp: le_divide_eq tendsto_const)
hoelzl@51642
   166
qed fact
hoelzl@51642
   167
hoelzl@56181
   168
lemma has_derivative_subset: "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
hoelzl@56181
   169
  by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)
hoelzl@51642
   170
hoelzl@51642
   171
subsection {* Continuity *}
hoelzl@51642
   172
hoelzl@56181
   173
lemma has_derivative_continuous:
hoelzl@56181
   174
  assumes f: "(f has_derivative f') (at x within s)"
hoelzl@51642
   175
  shows "continuous (at x within s) f"
hoelzl@51642
   176
proof -
hoelzl@56181
   177
  from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
hoelzl@51642
   178
  note F.tendsto[tendsto_intros]
hoelzl@51642
   179
  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
hoelzl@51642
   180
  have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
hoelzl@56181
   181
    using f unfolding has_derivative_iff_norm by blast
hoelzl@51642
   182
  then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
hoelzl@51642
   183
    by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
hoelzl@51642
   184
  also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
hoelzl@51642
   185
    by (intro filterlim_cong) (simp_all add: eventually_at_filter)
hoelzl@51642
   186
  finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
hoelzl@51642
   187
    by (rule tendsto_norm_zero_cancel)
hoelzl@51642
   188
  then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
hoelzl@51642
   189
    by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
hoelzl@51642
   190
  then have "?L (\<lambda>y. f y - f x)"
hoelzl@51642
   191
    by simp
hoelzl@51642
   192
  from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
hoelzl@51642
   193
    by (simp add: continuous_within)
hoelzl@51642
   194
qed
hoelzl@51642
   195
hoelzl@51642
   196
subsection {* Composition *}
hoelzl@51642
   197
hoelzl@51642
   198
lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f ---> y) (at x within s) \<longleftrightarrow> (f ---> y) (inf (nhds x) (principal s))"
hoelzl@51642
   199
  unfolding tendsto_def eventually_inf_principal eventually_at_filter
hoelzl@51642
   200
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@51642
   201
hoelzl@56181
   202
lemma has_derivative_in_compose:
hoelzl@56181
   203
  assumes f: "(f has_derivative f') (at x within s)"
hoelzl@56181
   204
  assumes g: "(g has_derivative g') (at (f x) within (f`s))"
hoelzl@56181
   205
  shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
hoelzl@51642
   206
proof -
hoelzl@56181
   207
  from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
hoelzl@56181
   208
  from g interpret G: bounded_linear g' by (rule has_derivative_bounded_linear)
hoelzl@51642
   209
  from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF" by fast
hoelzl@51642
   210
  from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast
hoelzl@51642
   211
  note G.tendsto[tendsto_intros]
hoelzl@51642
   212
hoelzl@51642
   213
  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
hoelzl@51642
   214
  let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
hoelzl@51642
   215
  let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
hoelzl@51642
   216
  let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
hoelzl@51642
   217
  def Nf \<equiv> "?N f f' x"
hoelzl@51642
   218
  def Ng \<equiv> "\<lambda>y. ?N g g' (f x) (f y)"
hoelzl@51642
   219
hoelzl@51642
   220
  show ?thesis
hoelzl@56181
   221
  proof (rule has_derivativeI_sandwich[of 1])
hoelzl@51642
   222
    show "bounded_linear (\<lambda>x. g' (f' x))"
hoelzl@56181
   223
      using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
hoelzl@51642
   224
  next
hoelzl@51642
   225
    fix y::'a assume neq: "y \<noteq> x"
hoelzl@51642
   226
    have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
hoelzl@51642
   227
      by (simp add: G.diff G.add field_simps)
hoelzl@51642
   228
    also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
hoelzl@51642
   229
      by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def)
hoelzl@51642
   230
    also have "\<dots> \<le> Nf y * kG + Ng y * (Nf y + kF)"
hoelzl@51642
   231
    proof (intro add_mono mult_left_mono)
hoelzl@51642
   232
      have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"
hoelzl@51642
   233
        by simp
hoelzl@51642
   234
      also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
hoelzl@51642
   235
        by (rule norm_triangle_ineq)
hoelzl@51642
   236
      also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
hoelzl@51642
   237
        using kF by (intro add_mono) simp
hoelzl@51642
   238
      finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
hoelzl@51642
   239
        by (simp add: neq Nf_def field_simps)
hoelzl@51642
   240
    qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
hoelzl@51642
   241
    finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
hoelzl@51642
   242
  next
hoelzl@51642
   243
    have [tendsto_intros]: "?L Nf"
hoelzl@56181
   244
      using f unfolding has_derivative_iff_norm Nf_def ..
hoelzl@51642
   245
    from f have "(f ---> f x) (at x within s)"
hoelzl@56181
   246
      by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
hoelzl@51642
   247
    then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
hoelzl@51642
   248
      unfolding filterlim_def
hoelzl@51642
   249
      by (simp add: eventually_filtermap eventually_at_filter le_principal)
hoelzl@51642
   250
hoelzl@51642
   251
    have "((?N g  g' (f x)) ---> 0) (at (f x) within f`s)"
hoelzl@56181
   252
      using g unfolding has_derivative_iff_norm ..
hoelzl@51642
   253
    then have g': "((?N g  g' (f x)) ---> 0) (inf (nhds (f x)) (principal (f`s)))"
hoelzl@51642
   254
      by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
hoelzl@51642
   255
hoelzl@51642
   256
    have [tendsto_intros]: "?L Ng"
hoelzl@51642
   257
      unfolding Ng_def by (rule filterlim_compose[OF g' f'])
hoelzl@51642
   258
    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) ---> 0) (at x within s)"
hoelzl@51642
   259
      by (intro tendsto_eq_intros) auto
hoelzl@51642
   260
  qed simp
hoelzl@51642
   261
qed
hoelzl@51642
   262
hoelzl@56181
   263
lemma has_derivative_compose:
hoelzl@56181
   264
  "(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
hoelzl@56181
   265
  ((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
hoelzl@56181
   266
  by (blast intro: has_derivative_in_compose has_derivative_subset)
hoelzl@51642
   267
hoelzl@51642
   268
lemma (in bounded_bilinear) FDERIV:
hoelzl@56181
   269
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
hoelzl@56181
   270
  shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
hoelzl@51642
   271
proof -
hoelzl@56181
   272
  from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
hoelzl@51642
   273
  obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
hoelzl@51642
   274
hoelzl@51642
   275
  from pos_bounded obtain K where K: "0 < K" and norm_prod:
hoelzl@51642
   276
    "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
hoelzl@51642
   277
  let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
hoelzl@51642
   278
  let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
hoelzl@51642
   279
  def Ng =="?N g g'" and Nf =="?N f f'"
hoelzl@51642
   280
hoelzl@51642
   281
  let ?fun1 = "\<lambda>y. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)"
hoelzl@51642
   282
  let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
hoelzl@51642
   283
  let ?F = "at x within s"
huffman@21164
   284
hoelzl@51642
   285
  show ?thesis
hoelzl@56181
   286
  proof (rule has_derivativeI_sandwich[of 1])
hoelzl@51642
   287
    show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
hoelzl@51642
   288
      by (intro bounded_linear_add
hoelzl@51642
   289
        bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
hoelzl@56181
   290
        has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
hoelzl@51642
   291
  next
hoelzl@51642
   292
    from g have "(g ---> g x) ?F"
hoelzl@56181
   293
      by (intro continuous_within[THEN iffD1] has_derivative_continuous)
hoelzl@51642
   294
    moreover from f g have "(Nf ---> 0) ?F" "(Ng ---> 0) ?F"
hoelzl@56181
   295
      by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
hoelzl@51642
   296
    ultimately have "(?fun2 ---> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
hoelzl@51642
   297
      by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
hoelzl@51642
   298
    then show "(?fun2 ---> 0) ?F"
hoelzl@51642
   299
      by simp
hoelzl@51642
   300
  next
hoelzl@51642
   301
    fix y::'d assume "y \<noteq> x"
hoelzl@51642
   302
    have "?fun1 y = norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"
hoelzl@51642
   303
      by (simp add: diff_left diff_right add_left add_right field_simps)
hoelzl@51642
   304
    also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
hoelzl@51642
   305
        norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
hoelzl@51642
   306
      by (intro divide_right_mono mult_mono'
hoelzl@51642
   307
                order_trans [OF norm_triangle_ineq add_mono]
hoelzl@51642
   308
                order_trans [OF norm_prod mult_right_mono]
hoelzl@51642
   309
                mult_nonneg_nonneg order_refl norm_ge_zero norm_F
hoelzl@51642
   310
                K [THEN order_less_imp_le])
hoelzl@51642
   311
    also have "\<dots> = ?fun2 y"
hoelzl@51642
   312
      by (simp add: add_divide_distrib Ng_def Nf_def)
hoelzl@51642
   313
    finally show "?fun1 y \<le> ?fun2 y" .
hoelzl@51642
   314
  qed simp
hoelzl@51642
   315
qed
hoelzl@51642
   316
hoelzl@56181
   317
lemmas has_derivative_mult[simp, has_derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
hoelzl@56181
   318
lemmas has_derivative_scaleR[simp, has_derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
hoelzl@51642
   319
hoelzl@56181
   320
lemma has_derivative_setprod[simp, has_derivative_intros]:
hoelzl@51642
   321
  fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   322
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within s)"
hoelzl@56181
   323
  shows "((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within s)"
hoelzl@51642
   324
proof cases
hoelzl@51642
   325
  assume "finite I" from this f show ?thesis
hoelzl@51642
   326
  proof induct
hoelzl@51642
   327
    case (insert i I)
hoelzl@51642
   328
    let ?P = "\<lambda>y. f i x * (\<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x)) + (f' i y) * (\<Prod>i\<in>I. f i x)"
hoelzl@56181
   329
    have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"
hoelzl@56181
   330
      using insert by (intro has_derivative_mult) auto
hoelzl@51642
   331
    also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
hoelzl@51642
   332
      using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum_cong)
hoelzl@51642
   333
    finally show ?case
hoelzl@51642
   334
      using insert by simp
hoelzl@51642
   335
  qed simp  
hoelzl@51642
   336
qed simp
hoelzl@51642
   337
hoelzl@56181
   338
lemma has_derivative_power[simp, has_derivative_intros]:
hoelzl@51642
   339
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   340
  assumes f: "(f has_derivative f') (at x within s)"
hoelzl@56181
   341
  shows "((\<lambda>x. f x^n) has_derivative (\<lambda>y. of_nat n * f' y * f x^(n - 1))) (at x within s)"
hoelzl@56181
   342
  using has_derivative_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)
hoelzl@51642
   343
hoelzl@56181
   344
lemma has_derivative_inverse':
hoelzl@51642
   345
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51642
   346
  assumes x: "x \<noteq> 0"
hoelzl@56181
   347
  shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"
hoelzl@56181
   348
        (is "(?inv has_derivative ?f) _")
hoelzl@56181
   349
proof (rule has_derivativeI_sandwich)
hoelzl@51642
   350
  show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
hoelzl@51642
   351
    apply (rule bounded_linear_minus)
hoelzl@51642
   352
    apply (rule bounded_linear_mult_const)
hoelzl@51642
   353
    apply (rule bounded_linear_const_mult)
hoelzl@51642
   354
    apply (rule bounded_linear_ident)
hoelzl@51642
   355
    done
hoelzl@51642
   356
next
hoelzl@51642
   357
  show "0 < norm x" using x by simp
hoelzl@51642
   358
next
hoelzl@51642
   359
  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) ---> 0) (at x within s)"
hoelzl@51642
   360
    apply (rule tendsto_mult_left_zero)
hoelzl@51642
   361
    apply (rule tendsto_norm_zero)
hoelzl@51642
   362
    apply (rule LIM_zero)
hoelzl@51642
   363
    apply (rule tendsto_inverse)
hoelzl@51642
   364
    apply (rule tendsto_ident_at)
hoelzl@51642
   365
    apply (rule x)
hoelzl@51642
   366
    done
hoelzl@51642
   367
next
hoelzl@51642
   368
  fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"
hoelzl@51642
   369
  then have "y \<noteq> 0"
hoelzl@51642
   370
    by (auto simp: norm_conv_dist dist_commute)
hoelzl@51642
   371
  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
hoelzl@51642
   372
    apply (subst inverse_diff_inverse [OF `y \<noteq> 0` x])
hoelzl@51642
   373
    apply (subst minus_diff_minus)
hoelzl@51642
   374
    apply (subst norm_minus_cancel)
hoelzl@51642
   375
    apply (simp add: left_diff_distrib)
hoelzl@51642
   376
    done
hoelzl@51642
   377
  also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
hoelzl@51642
   378
    apply (rule divide_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   379
    apply (rule order_trans [OF norm_mult_ineq])
hoelzl@51642
   380
    apply (rule mult_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   381
    apply (rule norm_mult_ineq)
hoelzl@51642
   382
    done
hoelzl@51642
   383
  also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
hoelzl@51642
   384
    by simp
hoelzl@51642
   385
  finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
hoelzl@51642
   386
      norm (?inv y - ?inv x) * norm (?inv x)" .
hoelzl@51642
   387
qed
hoelzl@51642
   388
hoelzl@56181
   389
lemma has_derivative_inverse[simp, has_derivative_intros]:
hoelzl@51642
   390
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@56181
   391
  assumes x:  "f x \<noteq> 0" and f: "(f has_derivative f') (at x within s)"
hoelzl@56181
   392
  shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within s)"
hoelzl@56181
   393
  using has_derivative_compose[OF f has_derivative_inverse', OF x] .
hoelzl@51642
   394
hoelzl@56181
   395
lemma has_derivative_divide[simp, has_derivative_intros]:
hoelzl@51642
   396
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@56181
   397
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" 
lp15@55967
   398
  assumes x: "g x \<noteq> 0"
hoelzl@56181
   399
  shows "((\<lambda>x. f x / g x) has_derivative
hoelzl@56181
   400
                (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"
hoelzl@56181
   401
  using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
lp15@55967
   402
  by (simp add: divide_inverse field_simps)
lp15@55967
   403
lp15@55967
   404
text{*Conventional form requires mult-AC laws. Types real and complex only.*}
hoelzl@56181
   405
hoelzl@56181
   406
lemma has_derivative_divide'[has_derivative_intros]: 
lp15@55967
   407
  fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
hoelzl@56181
   408
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" and x: "g x \<noteq> 0"
hoelzl@56181
   409
  shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within s)"
lp15@55967
   410
proof -
lp15@55967
   411
  { fix h
lp15@55967
   412
    have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
lp15@55967
   413
          (f' h * g x - f x * g' h) / (g x * g x)"
lp15@55967
   414
      by (simp add: divide_inverse field_simps nonzero_inverse_mult_distrib x)
lp15@55967
   415
   }
lp15@55967
   416
  then show ?thesis
hoelzl@56181
   417
    using has_derivative_divide [OF f g] x
lp15@55967
   418
    by simp
lp15@55967
   419
qed
hoelzl@51642
   420
hoelzl@51642
   421
subsection {* Uniqueness *}
hoelzl@51642
   422
hoelzl@51642
   423
text {*
hoelzl@51642
   424
hoelzl@56181
   425
This can not generally shown for @{const has_derivative}, as we need to approach the point from
hoelzl@51642
   426
all directions. There is a proof in @{text Multivariate_Analysis} for @{text euclidean_space}.
hoelzl@51642
   427
hoelzl@51642
   428
*}
hoelzl@51642
   429
hoelzl@56181
   430
lemma has_derivative_zero_unique:
hoelzl@56181
   431
  assumes "((\<lambda>x. 0) has_derivative F) (at x)" shows "F = (\<lambda>h. 0)"
hoelzl@51642
   432
proof -
hoelzl@51642
   433
  interpret F: bounded_linear F
hoelzl@56181
   434
    using assms by (rule has_derivative_bounded_linear)
hoelzl@51642
   435
  let ?r = "\<lambda>h. norm (F h) / norm h"
hoelzl@51642
   436
  have *: "?r -- 0 --> 0"
hoelzl@56181
   437
    using assms unfolding has_derivative_at by simp
hoelzl@51642
   438
  show "F = (\<lambda>h. 0)"
hoelzl@51642
   439
  proof
hoelzl@51642
   440
    fix h show "F h = 0"
hoelzl@51642
   441
    proof (rule ccontr)
wenzelm@53374
   442
      assume **: "F h \<noteq> 0"
wenzelm@53374
   443
      then have h: "h \<noteq> 0"
hoelzl@51642
   444
        by (clarsimp simp add: F.zero)
wenzelm@53374
   445
      with ** have "0 < ?r h"
hoelzl@51642
   446
        by (simp add: divide_pos_pos)
hoelzl@51642
   447
      from LIM_D [OF * this] obtain s where s: "0 < s"
hoelzl@51642
   448
        and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
hoelzl@51642
   449
      from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
hoelzl@51642
   450
      let ?x = "scaleR (t / norm h) h"
hoelzl@51642
   451
      have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
hoelzl@51642
   452
      hence "?r ?x < ?r h" by (rule r)
hoelzl@51642
   453
      thus "False" using t h by (simp add: F.scaleR)
hoelzl@51642
   454
    qed
hoelzl@51642
   455
  qed
hoelzl@51642
   456
qed
hoelzl@51642
   457
hoelzl@56181
   458
lemma has_derivative_unique:
hoelzl@56181
   459
  assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'"
hoelzl@51642
   460
proof -
hoelzl@56181
   461
  have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
hoelzl@56181
   462
    using has_derivative_diff [OF assms] by simp
hoelzl@51642
   463
  hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
hoelzl@56181
   464
    by (rule has_derivative_zero_unique)
hoelzl@51642
   465
  thus "F = F'"
hoelzl@51642
   466
    unfolding fun_eq_iff right_minus_eq .
hoelzl@51642
   467
qed
hoelzl@51642
   468
hoelzl@51642
   469
subsection {* Differentiability predicate *}
hoelzl@51642
   470
hoelzl@56181
   471
definition
hoelzl@56181
   472
  differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@56181
   473
  (infixr "differentiable" 30)
hoelzl@56181
   474
where
hoelzl@56181
   475
  "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
hoelzl@51642
   476
hoelzl@56181
   477
lemma differentiable_subset: "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
hoelzl@56181
   478
  unfolding differentiable_def by (blast intro: has_derivative_subset)
hoelzl@51642
   479
hoelzl@56181
   480
lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable F"
hoelzl@56181
   481
  unfolding differentiable_def by (blast intro: has_derivative_ident)
hoelzl@51642
   482
hoelzl@56181
   483
lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable F"
hoelzl@56181
   484
  unfolding differentiable_def by (blast intro: has_derivative_const)
hoelzl@51642
   485
hoelzl@51642
   486
lemma differentiable_in_compose:
hoelzl@56181
   487
  "f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
hoelzl@56181
   488
  unfolding differentiable_def by (blast intro: has_derivative_in_compose)
hoelzl@51642
   489
hoelzl@51642
   490
lemma differentiable_compose:
hoelzl@56181
   491
  "f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
hoelzl@51642
   492
  by (blast intro: differentiable_in_compose differentiable_subset)
hoelzl@51642
   493
hoelzl@51642
   494
lemma differentiable_sum [simp]:
hoelzl@56181
   495
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
hoelzl@56181
   496
  unfolding differentiable_def by (blast intro: has_derivative_add)
hoelzl@51642
   497
hoelzl@51642
   498
lemma differentiable_minus [simp]:
hoelzl@56181
   499
  "f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
hoelzl@56181
   500
  unfolding differentiable_def by (blast intro: has_derivative_minus)
hoelzl@51642
   501
hoelzl@51642
   502
lemma differentiable_diff [simp]:
hoelzl@56181
   503
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
hoelzl@56181
   504
  unfolding differentiable_def by (blast intro: has_derivative_diff)
hoelzl@51642
   505
hoelzl@51642
   506
lemma differentiable_mult [simp]:
hoelzl@51642
   507
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
hoelzl@56181
   508
  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x * g x) differentiable (at x within s)"
hoelzl@56181
   509
  unfolding differentiable_def by (blast intro: has_derivative_mult)
hoelzl@51642
   510
hoelzl@51642
   511
lemma differentiable_inverse [simp]:
hoelzl@51642
   512
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   513
  shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable (at x within s)"
hoelzl@56181
   514
  unfolding differentiable_def by (blast intro: has_derivative_inverse)
hoelzl@51642
   515
hoelzl@51642
   516
lemma differentiable_divide [simp]:
hoelzl@51642
   517
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   518
  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
hoelzl@51642
   519
  unfolding divide_inverse using assms by simp
hoelzl@51642
   520
hoelzl@51642
   521
lemma differentiable_power [simp]:
hoelzl@51642
   522
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   523
  shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
hoelzl@56181
   524
  unfolding differentiable_def by (blast intro: has_derivative_power)
hoelzl@51642
   525
hoelzl@51642
   526
lemma differentiable_scaleR [simp]:
hoelzl@56181
   527
  "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
hoelzl@56181
   528
  unfolding differentiable_def by (blast intro: has_derivative_scaleR)
hoelzl@51642
   529
hoelzl@51642
   530
definition 
hoelzl@56181
   531
  has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@56181
   532
  (infixl "(has'_field'_derivative)" 12)
hoelzl@51642
   533
where
hoelzl@56181
   534
  "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"
hoelzl@56181
   535
hoelzl@56181
   536
lemma has_derivative_imp_has_field_derivative:
hoelzl@56181
   537
  "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
hoelzl@56181
   538
  unfolding has_field_derivative_def 
hoelzl@56181
   539
  by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult_commute)
hoelzl@56181
   540
hoelzl@56181
   541
lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
hoelzl@56181
   542
  by (simp add: has_field_derivative_def)
hoelzl@51642
   543
hoelzl@56181
   544
abbreviation (input)
hoelzl@56181
   545
  deriv :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@56181
   546
  ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
hoelzl@51642
   547
where
hoelzl@56181
   548
  "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
hoelzl@51642
   549
hoelzl@56181
   550
abbreviation 
hoelzl@56181
   551
  has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
hoelzl@56181
   552
  (infixl "(has'_real'_derivative)" 12)
hoelzl@56181
   553
where
hoelzl@56181
   554
  "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
hoelzl@56181
   555
hoelzl@56181
   556
lemma real_differentiable_def:
hoelzl@56181
   557
  "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
hoelzl@51642
   558
proof safe
hoelzl@56181
   559
  assume "f differentiable at x within s"
hoelzl@56181
   560
  then obtain f' where *: "(f has_derivative f') (at x within s)"
hoelzl@56181
   561
    unfolding differentiable_def by auto
hoelzl@56181
   562
  then obtain c where "f' = (op * c)"
hoelzl@56181
   563
    by (metis real_bounded_linear has_derivative_bounded_linear mult_commute fun_eq_iff)
hoelzl@56181
   564
  with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181
   565
    unfolding has_field_derivative_def by auto
hoelzl@56181
   566
qed (auto simp: differentiable_def has_field_derivative_def)
hoelzl@51642
   567
hoelzl@56181
   568
lemma real_differentiableE [elim?]:
hoelzl@56181
   569
  assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)"
hoelzl@56181
   570
  using assms by (auto simp: real_differentiable_def)
hoelzl@51642
   571
hoelzl@56181
   572
lemma differentiableD: "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181
   573
  by (auto elim: real_differentiableE)
hoelzl@51642
   574
hoelzl@56181
   575
lemma differentiableI: "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"
hoelzl@56181
   576
  by (force simp add: real_differentiable_def)
hoelzl@51642
   577
hoelzl@51642
   578
lemma deriv_def:
hoelzl@51642
   579
  "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@56181
   580
  apply (simp add: has_field_derivative_def has_derivative_at bounded_linear_mult_right LIM_zero_iff[symmetric, of _ D])
hoelzl@51642
   581
  apply (subst (2) tendsto_norm_zero_iff[symmetric])
hoelzl@51642
   582
  apply (rule filterlim_cong)
hoelzl@51642
   583
  apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
hoelzl@51642
   584
  done
huffman@21164
   585
hoelzl@56181
   586
lemma mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"
hoelzl@56181
   587
  by (simp add: fun_eq_iff mult_commute)
huffman@21164
   588
hoelzl@56181
   589
subsection {* Derivatives *}
huffman@21164
   590
hoelzl@51642
   591
lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@51642
   592
  by (simp add: deriv_def)
huffman@21164
   593
hoelzl@56181
   594
lemma DERIV_const [simp]: "((\<lambda>x. k) has_field_derivative 0) (at x within s)"
hoelzl@56181
   595
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto
hoelzl@56181
   596
hoelzl@56181
   597
lemma DERIV_ident [simp]: "((\<lambda>x. x) has_field_derivative 1) (at x within s)"
hoelzl@56181
   598
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
huffman@21164
   599
hoelzl@56181
   600
lemma DERIV_add:
hoelzl@56181
   601
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   602
  ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
hoelzl@56181
   603
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
hoelzl@56181
   604
     (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
huffman@21164
   605
hoelzl@56181
   606
lemma DERIV_minus: "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
hoelzl@56181
   607
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus]) 
hoelzl@56181
   608
     (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
huffman@21164
   609
hoelzl@56181
   610
lemma DERIV_diff:
hoelzl@56181
   611
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   612
  ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
hoelzl@56181
   613
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_diff])
hoelzl@56181
   614
     (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
huffman@21164
   615
hoelzl@56181
   616
lemma DERIV_add_minus:
hoelzl@56181
   617
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   618
  ((\<lambda>x. f x + - g x) has_field_derivative D + - E) (at x within s)"
hoelzl@51642
   619
  by (simp only: DERIV_add DERIV_minus)
hoelzl@51642
   620
hoelzl@56181
   621
lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
hoelzl@56181
   622
  by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp
huffman@21164
   623
huffman@21164
   624
lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
hoelzl@51642
   625
  by (auto dest!: DERIV_continuous)
hoelzl@51642
   626
hoelzl@56181
   627
lemma DERIV_mult':
hoelzl@56181
   628
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   629
  ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
hoelzl@56181
   630
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181
   631
     (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
huffman@21164
   632
hoelzl@56181
   633
lemma DERIV_mult:
hoelzl@56181
   634
  "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   635
  ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
hoelzl@56181
   636
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181
   637
     (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   638
hoelzl@51642
   639
text {* Derivative of linear multiplication *}
huffman@21164
   640
hoelzl@51642
   641
lemma DERIV_cmult:
hoelzl@56181
   642
  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
hoelzl@51642
   643
  by (drule DERIV_mult' [OF DERIV_const], simp)
huffman@21164
   644
lp15@55967
   645
lemma DERIV_cmult_right:
hoelzl@56181
   646
  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
lp15@55967
   647
  using DERIV_cmult   by (force simp add: mult_ac)
lp15@55967
   648
hoelzl@56181
   649
lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
hoelzl@51642
   650
  by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
hoelzl@51642
   651
hoelzl@56181
   652
lemma DERIV_cdivide:
hoelzl@56181
   653
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
hoelzl@56181
   654
  using DERIV_cmult_right[of f D x s "1 / c"] by simp
huffman@21164
   655
huffman@21164
   656
lemma DERIV_unique:
hoelzl@51642
   657
  "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
hoelzl@50331
   658
  unfolding deriv_def by (rule LIM_unique) 
huffman@21164
   659
hoelzl@31880
   660
lemma DERIV_setsum:
hoelzl@56181
   661
  "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow> 
hoelzl@56181
   662
    ((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"
hoelzl@56181
   663
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_setsum])
hoelzl@56181
   664
     (auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   665
hoelzl@51642
   666
lemma DERIV_inverse':
hoelzl@56181
   667
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
hoelzl@56181
   668
  ((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"
hoelzl@56181
   669
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_inverse])
hoelzl@56181
   670
     (auto dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   671
hoelzl@51642
   672
text {* Power of @{text "-1"} *}
hoelzl@51642
   673
hoelzl@51642
   674
lemma DERIV_inverse:
hoelzl@56181
   675
  "x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
hoelzl@51642
   676
  by (drule DERIV_inverse' [OF DERIV_ident]) simp
hoelzl@51642
   677
hoelzl@51642
   678
text {* Derivative of inverse *}
hoelzl@51642
   679
hoelzl@51642
   680
lemma DERIV_inverse_fun:
hoelzl@56181
   681
  "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
hoelzl@56181
   682
  ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
hoelzl@51642
   683
  by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
hoelzl@51642
   684
hoelzl@51642
   685
text {* Derivative of quotient *}
hoelzl@51642
   686
hoelzl@51642
   687
lemma DERIV_divide:
hoelzl@56181
   688
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   689
  (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
hoelzl@56181
   690
  ((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
hoelzl@56181
   691
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
hoelzl@56181
   692
     (auto dest: has_field_derivative_imp_has_derivative simp: field_simps nonzero_inverse_mult_distrib divide_inverse)
hoelzl@51642
   693
hoelzl@51642
   694
lemma DERIV_quotient:
hoelzl@56181
   695
  "(f has_field_derivative d) (at x within s) \<Longrightarrow>
hoelzl@56181
   696
  (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> 
hoelzl@56181
   697
  ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
hoelzl@51642
   698
  by (drule (2) DERIV_divide) (simp add: mult_commute)
hoelzl@51642
   699
hoelzl@51642
   700
lemma DERIV_power_Suc:
hoelzl@56181
   701
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   702
  ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
hoelzl@56181
   703
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181
   704
     (auto simp: has_field_derivative_def)
hoelzl@51642
   705
hoelzl@51642
   706
lemma DERIV_power:
hoelzl@56181
   707
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   708
  ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
hoelzl@56181
   709
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181
   710
     (auto simp: has_field_derivative_def)
hoelzl@31880
   711
hoelzl@56181
   712
lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
hoelzl@51642
   713
  apply (cut_tac DERIV_power [OF DERIV_ident])
hoelzl@51642
   714
  apply (simp add: real_of_nat_def)
hoelzl@51642
   715
  done
hoelzl@51642
   716
hoelzl@56181
   717
lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> 
hoelzl@56181
   718
  ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
hoelzl@56181
   719
  using has_derivative_compose[of f "op * D" x s g "op * E"]
hoelzl@56181
   720
  unfolding has_field_derivative_def mult_commute_abs ac_simps .
hoelzl@51642
   721
hoelzl@56181
   722
corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   723
  ((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
lp15@55967
   724
  by (rule DERIV_chain')
lp15@55967
   725
hoelzl@51642
   726
text {* Standard version *}
hoelzl@51642
   727
hoelzl@56181
   728
lemma DERIV_chain:
hoelzl@56181
   729
  "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow> 
hoelzl@56181
   730
  (f o g has_field_derivative Da * Db) (at x within s)"
hoelzl@51642
   731
  by (drule (1) DERIV_chain', simp add: o_def mult_commute)
hoelzl@51642
   732
lp15@55967
   733
lemma DERIV_image_chain: 
hoelzl@56181
   734
  "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   735
  (f o g has_field_derivative Da * Db) (at x within s)"
hoelzl@56181
   736
  using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]
hoelzl@56181
   737
  by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
lp15@55967
   738
lp15@55967
   739
(*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
lp15@55967
   740
lemma DERIV_chain_s:
lp15@55967
   741
  assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
lp15@55967
   742
      and "DERIV f x :> f'" 
lp15@55967
   743
      and "f x \<in> s"
lp15@55967
   744
    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
lp15@55967
   745
  by (metis (full_types) DERIV_chain' mult_commute assms)
lp15@55967
   746
lp15@55967
   747
lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
lp15@55967
   748
  assumes "(\<And>x. DERIV g x :> g'(x))"
lp15@55967
   749
      and "DERIV f x :> f'" 
lp15@55967
   750
    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
lp15@55967
   751
  by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
lp15@55967
   752
hoelzl@51642
   753
hoelzl@51642
   754
subsubsection {* @{text "DERIV_intros"} *}
hoelzl@51642
   755
hoelzl@51642
   756
ML {*
hoelzl@51642
   757
structure Deriv_Intros = Named_Thms
hoelzl@51642
   758
(
hoelzl@51642
   759
  val name = @{binding DERIV_intros}
hoelzl@51642
   760
  val description = "DERIV introduction rules"
hoelzl@51642
   761
)
hoelzl@51642
   762
*}
hoelzl@51642
   763
hoelzl@51642
   764
setup Deriv_Intros.setup
hoelzl@51642
   765
hoelzl@56181
   766
lemma DERIV_cong: "(f has_field_derivative X) (at x within s) \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) (at x within s)"
hoelzl@51642
   767
  by simp
hoelzl@51642
   768
hoelzl@51642
   769
declare
hoelzl@51642
   770
  DERIV_const[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   771
  DERIV_ident[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   772
  DERIV_add[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   773
  DERIV_minus[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   774
  DERIV_mult[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   775
  DERIV_diff[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   776
  DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   777
  DERIV_divide[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   778
  DERIV_power[where 'a=real, THEN DERIV_cong,
hoelzl@51642
   779
              unfolded real_of_nat_def[symmetric], DERIV_intros]
hoelzl@51642
   780
  DERIV_setsum[THEN DERIV_cong, DERIV_intros]
huffman@21164
   781
huffman@21164
   782
text{*Alternative definition for differentiability*}
huffman@21164
   783
huffman@21164
   784
lemma DERIV_LIM_iff:
huffman@31338
   785
  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
huffman@21784
   786
     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
huffman@21164
   787
      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
huffman@21164
   788
apply (rule iffI)
huffman@21164
   789
apply (drule_tac k="- a" in LIM_offset)
haftmann@54230
   790
apply simp
huffman@21164
   791
apply (drule_tac k="a" in LIM_offset)
huffman@21164
   792
apply (simp add: add_commute)
huffman@21164
   793
done
huffman@21164
   794
hoelzl@51642
   795
lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"
haftmann@54230
   796
  by (simp add: deriv_def DERIV_LIM_iff)
huffman@21164
   797
hoelzl@51642
   798
lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
hoelzl@51642
   799
    DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
hoelzl@51642
   800
  unfolding DERIV_iff2
hoelzl@51642
   801
proof (rule filterlim_cong)
wenzelm@53374
   802
  assume *: "eventually (\<lambda>x. f x = g x) (nhds x)"
wenzelm@53374
   803
  moreover from * have "f x = g x" by (auto simp: eventually_nhds)
hoelzl@51642
   804
  moreover assume "x = y" "u = v"
hoelzl@51642
   805
  ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
hoelzl@51642
   806
    by (auto simp: eventually_at_filter elim: eventually_elim1)
hoelzl@51642
   807
qed simp_all
huffman@21164
   808
hoelzl@51642
   809
lemma DERIV_shift:
hoelzl@51642
   810
  "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
hoelzl@56181
   811
  by (simp add: deriv_def field_simps)
huffman@21164
   812
hoelzl@51642
   813
lemma DERIV_mirror:
hoelzl@51642
   814
  "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
hoelzl@51642
   815
  by (simp add: deriv_def filterlim_at_split filterlim_at_left_to_right
hoelzl@51642
   816
                tendsto_minus_cancel_left field_simps conj_commute)
huffman@21164
   817
huffman@29975
   818
text {* Caratheodory formulation of derivative at a point *}
huffman@21164
   819
lp15@55970
   820
lemma CARAT_DERIV: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)
hoelzl@51642
   821
  "(DERIV f x :> l) \<longleftrightarrow> (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
huffman@21164
   822
      (is "?lhs = ?rhs")
huffman@21164
   823
proof
huffman@21164
   824
  assume der: "DERIV f x :> l"
huffman@21784
   825
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
huffman@21164
   826
  proof (intro exI conjI)
huffman@21784
   827
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
nipkow@23413
   828
    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
huffman@21164
   829
    show "isCont ?g x" using der
hoelzl@56181
   830
      by (simp add: isCont_iff deriv_def cong: LIM_equal [rule_format])
huffman@21164
   831
    show "?g x = l" by simp
huffman@21164
   832
  qed
huffman@21164
   833
next
huffman@21164
   834
  assume "?rhs"
huffman@21164
   835
  then obtain g where
huffman@21784
   836
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
huffman@21164
   837
  thus "(DERIV f x :> l)"
hoelzl@56181
   838
     by (auto simp add: isCont_iff deriv_def cong: LIM_cong)
huffman@21164
   839
qed
huffman@21164
   840
wenzelm@31899
   841
text {*
wenzelm@31899
   842
 Let's do the standard proof, though theorem
wenzelm@31899
   843
 @{text "LIM_mult2"} follows from a NS proof
wenzelm@31899
   844
*}
huffman@21164
   845
huffman@29975
   846
subsection {* Local extrema *}
huffman@29975
   847
huffman@21164
   848
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
   849
paulson@33654
   850
lemma DERIV_pos_inc_right:
huffman@21164
   851
  fixes f :: "real => real"
huffman@21164
   852
  assumes der: "DERIV f x :> l"
huffman@21164
   853
      and l:   "0 < l"
huffman@21164
   854
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
   855
proof -
huffman@21164
   856
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
   857
  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)"
haftmann@54230
   858
    by simp
huffman@21164
   859
  then obtain s
huffman@21164
   860
        where s:   "0 < s"
huffman@21164
   861
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
   862
    by auto
huffman@21164
   863
  thus ?thesis
huffman@21164
   864
  proof (intro exI conjI strip)
huffman@23441
   865
    show "0<s" using s .
huffman@21164
   866
    fix h::real
huffman@21164
   867
    assume "0 < h" "h < s"
huffman@21164
   868
    with all [of h] show "f x < f (x+h)"
haftmann@54230
   869
    proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)
huffman@21164
   870
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
   871
      with l
huffman@21164
   872
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
   873
      thus "f x < f (x+h)"
huffman@21164
   874
  by (simp add: pos_less_divide_eq h)
huffman@21164
   875
    qed
huffman@21164
   876
  qed
huffman@21164
   877
qed
huffman@21164
   878
paulson@33654
   879
lemma DERIV_neg_dec_left:
huffman@21164
   880
  fixes f :: "real => real"
huffman@21164
   881
  assumes der: "DERIV f x :> l"
huffman@21164
   882
      and l:   "l < 0"
huffman@21164
   883
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
   884
proof -
huffman@21164
   885
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
   886
  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)"
haftmann@54230
   887
    by simp
huffman@21164
   888
  then obtain s
huffman@21164
   889
        where s:   "0 < s"
huffman@21164
   890
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
   891
    by auto
huffman@21164
   892
  thus ?thesis
huffman@21164
   893
  proof (intro exI conjI strip)
huffman@23441
   894
    show "0<s" using s .
huffman@21164
   895
    fix h::real
huffman@21164
   896
    assume "0 < h" "h < s"
huffman@21164
   897
    with all [of "-h"] show "f x < f (x-h)"
haftmann@54230
   898
    proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)
huffman@21164
   899
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
   900
      with l
huffman@21164
   901
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
   902
      thus "f x < f (x-h)"
huffman@21164
   903
  by (simp add: pos_less_divide_eq h)
huffman@21164
   904
    qed
huffman@21164
   905
  qed
huffman@21164
   906
qed
huffman@21164
   907
paulson@33654
   908
lemma DERIV_pos_inc_left:
paulson@33654
   909
  fixes f :: "real => real"
paulson@33654
   910
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
hoelzl@56181
   911
  apply (rule DERIV_neg_dec_left [of "%x. - f x" "-l" x, simplified])
hoelzl@41368
   912
  apply (auto simp add: DERIV_minus)
paulson@33654
   913
  done
paulson@33654
   914
paulson@33654
   915
lemma DERIV_neg_dec_right:
paulson@33654
   916
  fixes f :: "real => real"
paulson@33654
   917
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
hoelzl@56181
   918
  apply (rule DERIV_pos_inc_right [of "%x. - f x" "-l" x, simplified])
hoelzl@41368
   919
  apply (auto simp add: DERIV_minus)
paulson@33654
   920
  done
paulson@33654
   921
huffman@21164
   922
lemma DERIV_local_max:
huffman@21164
   923
  fixes f :: "real => real"
huffman@21164
   924
  assumes der: "DERIV f x :> l"
huffman@21164
   925
      and d:   "0 < d"
huffman@21164
   926
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
   927
  shows "l = 0"
huffman@21164
   928
proof (cases rule: linorder_cases [of l 0])
huffman@23441
   929
  case equal thus ?thesis .
huffman@21164
   930
next
huffman@21164
   931
  case less
paulson@33654
   932
  from DERIV_neg_dec_left [OF der less]
huffman@21164
   933
  obtain d' where d': "0 < d'"
huffman@21164
   934
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
   935
  from real_lbound_gt_zero [OF d d']
huffman@21164
   936
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   937
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
   938
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   939
next
huffman@21164
   940
  case greater
paulson@33654
   941
  from DERIV_pos_inc_right [OF der greater]
huffman@21164
   942
  obtain d' where d': "0 < d'"
huffman@21164
   943
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
   944
  from real_lbound_gt_zero [OF d d']
huffman@21164
   945
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   946
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
   947
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   948
qed
huffman@21164
   949
huffman@21164
   950
huffman@21164
   951
text{*Similar theorem for a local minimum*}
huffman@21164
   952
lemma DERIV_local_min:
huffman@21164
   953
  fixes f :: "real => real"
huffman@21164
   954
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
   955
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
   956
huffman@21164
   957
huffman@21164
   958
text{*In particular, if a function is locally flat*}
huffman@21164
   959
lemma DERIV_local_const:
huffman@21164
   960
  fixes f :: "real => real"
huffman@21164
   961
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
   962
by (auto dest!: DERIV_local_max)
huffman@21164
   963
huffman@29975
   964
huffman@29975
   965
subsection {* Rolle's Theorem *}
huffman@29975
   966
huffman@21164
   967
text{*Lemma about introducing open ball in open interval*}
huffman@21164
   968
lemma lemma_interval_lt:
huffman@21164
   969
     "[| a < x;  x < b |]
huffman@21164
   970
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
chaieb@27668
   971
huffman@22998
   972
apply (simp add: abs_less_iff)
huffman@21164
   973
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
   974
apply (rule_tac x = "x-a" in exI)
huffman@21164
   975
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
   976
done
huffman@21164
   977
huffman@21164
   978
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
   979
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
   980
apply (drule lemma_interval_lt, auto)
huffman@44921
   981
apply force
huffman@21164
   982
done
huffman@21164
   983
huffman@21164
   984
text{*Rolle's Theorem.
huffman@21164
   985
   If @{term f} is defined and continuous on the closed interval
huffman@21164
   986
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
   987
   and @{term "f(a) = f(b)"},
huffman@21164
   988
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
   989
theorem Rolle:
huffman@21164
   990
  assumes lt: "a < b"
huffman@21164
   991
      and eq: "f(a) = f(b)"
huffman@21164
   992
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
hoelzl@56181
   993
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
huffman@21784
   994
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164
   995
proof -
huffman@21164
   996
  have le: "a \<le> b" using lt by simp
huffman@21164
   997
  from isCont_eq_Ub [OF le con]
huffman@21164
   998
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
   999
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
  1000
    by blast
huffman@21164
  1001
  from isCont_eq_Lb [OF le con]
huffman@21164
  1002
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
  1003
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
  1004
    by blast
huffman@21164
  1005
  show ?thesis
huffman@21164
  1006
  proof cases
huffman@21164
  1007
    assume axb: "a < x & x < b"
huffman@21164
  1008
        --{*@{term f} attains its maximum within the interval*}
chaieb@27668
  1009
    hence ax: "a<x" and xb: "x<b" by arith + 
huffman@21164
  1010
    from lemma_interval [OF ax xb]
huffman@21164
  1011
    obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164
  1012
      by blast
huffman@21164
  1013
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
  1014
      by blast
huffman@21164
  1015
    from differentiableD [OF dif [OF axb]]
huffman@21164
  1016
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1017
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
  1018
        --{*the derivative at a local maximum is zero*}
huffman@21164
  1019
    thus ?thesis using ax xb der by auto
huffman@21164
  1020
  next
huffman@21164
  1021
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
  1022
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
  1023
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
  1024
    show ?thesis
huffman@21164
  1025
    proof cases
huffman@21164
  1026
      assume ax'b: "a < x' & x' < b"
huffman@21164
  1027
        --{*@{term f} attains its minimum within the interval*}
chaieb@27668
  1028
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
huffman@21164
  1029
      from lemma_interval [OF ax' x'b]
huffman@21164
  1030
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164
  1031
  by blast
huffman@21164
  1032
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
  1033
  by blast
huffman@21164
  1034
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
  1035
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
  1036
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
  1037
        --{*the derivative at a local minimum is zero*}
huffman@21164
  1038
      thus ?thesis using ax' x'b der by auto
huffman@21164
  1039
    next
huffman@21164
  1040
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
  1041
        --{*@{term f} is constant througout the interval*}
huffman@21164
  1042
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
  1043
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
  1044
      from dense [OF lt]
huffman@21164
  1045
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
  1046
      from lemma_interval [OF ar rb]
huffman@21164
  1047
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164
  1048
  by blast
huffman@21164
  1049
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
  1050
      proof (clarify)
huffman@21164
  1051
        fix z::real
huffman@21164
  1052
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
  1053
        show "f z = f b"
huffman@21164
  1054
        proof (rule order_antisym)
huffman@21164
  1055
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
  1056
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
  1057
        qed
huffman@21164
  1058
      qed
huffman@21164
  1059
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
  1060
      proof (intro strip)
huffman@21164
  1061
        fix y::real
huffman@21164
  1062
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
  1063
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
  1064
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
  1065
      qed
huffman@21164
  1066
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
  1067
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
  1068
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
  1069
        --{*the derivative of a constant function is zero*}
huffman@21164
  1070
      thus ?thesis using ar rb der by auto
huffman@21164
  1071
    qed
huffman@21164
  1072
  qed
huffman@21164
  1073
qed
huffman@21164
  1074
huffman@21164
  1075
huffman@21164
  1076
subsection{*Mean Value Theorem*}
huffman@21164
  1077
huffman@21164
  1078
lemma lemma_MVT:
huffman@21164
  1079
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
hoelzl@51481
  1080
  by (cases "a = b") (simp_all add: field_simps)
huffman@21164
  1081
huffman@21164
  1082
theorem MVT:
huffman@21164
  1083
  assumes lt:  "a < b"
huffman@21164
  1084
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
hoelzl@56181
  1085
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
huffman@21784
  1086
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164
  1087
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
  1088
proof -
huffman@21164
  1089
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@44233
  1090
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
huffman@44233
  1091
    using con by (fast intro: isCont_intros)
hoelzl@56181
  1092
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
huffman@21164
  1093
  proof (clarify)
huffman@21164
  1094
    fix x::real
huffman@21164
  1095
    assume ax: "a < x" and xb: "x < b"
huffman@21164
  1096
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
  1097
    obtain l where der: "DERIV f x :> l" ..
hoelzl@56181
  1098
    show "?F differentiable (at x)"
huffman@21164
  1099
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
  1100
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1101
  qed
huffman@21164
  1102
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1103
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1104
    by blast
huffman@21164
  1105
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1106
    by (rule DERIV_cmult_Id)
huffman@21164
  1107
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1108
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1109
    by (rule DERIV_add [OF der])
huffman@21164
  1110
  show ?thesis
huffman@21164
  1111
  proof (intro exI conjI)
huffman@23441
  1112
    show "a < z" using az .
huffman@23441
  1113
    show "z < b" using zb .
huffman@21164
  1114
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1115
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1116
  qed
huffman@21164
  1117
qed
huffman@21164
  1118
hoelzl@29803
  1119
lemma MVT2:
hoelzl@29803
  1120
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
hoelzl@29803
  1121
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
hoelzl@29803
  1122
apply (drule MVT)
hoelzl@29803
  1123
apply (blast intro: DERIV_isCont)
hoelzl@56181
  1124
apply (force dest: order_less_imp_le simp add: real_differentiable_def)
hoelzl@29803
  1125
apply (blast dest: DERIV_unique order_less_imp_le)
hoelzl@29803
  1126
done
hoelzl@29803
  1127
huffman@21164
  1128
huffman@21164
  1129
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1130
huffman@21164
  1131
lemma DERIV_isconst_end:
huffman@21164
  1132
  fixes f :: "real => real"
huffman@21164
  1133
  shows "[| a < b;
huffman@21164
  1134
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1135
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1136
        ==> f b = f a"
huffman@21164
  1137
apply (drule MVT, assumption)
huffman@21164
  1138
apply (blast intro: differentiableI)
huffman@21164
  1139
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1140
done
huffman@21164
  1141
huffman@21164
  1142
lemma DERIV_isconst1:
huffman@21164
  1143
  fixes f :: "real => real"
huffman@21164
  1144
  shows "[| a < b;
huffman@21164
  1145
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1146
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1147
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1148
apply safe
huffman@21164
  1149
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1150
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1151
done
huffman@21164
  1152
huffman@21164
  1153
lemma DERIV_isconst2:
huffman@21164
  1154
  fixes f :: "real => real"
huffman@21164
  1155
  shows "[| a < b;
huffman@21164
  1156
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1157
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1158
         a \<le> x; x \<le> b |]
huffman@21164
  1159
        ==> f x = f a"
huffman@21164
  1160
apply (blast dest: DERIV_isconst1)
huffman@21164
  1161
done
huffman@21164
  1162
hoelzl@29803
  1163
lemma DERIV_isconst3: fixes a b x y :: real
hoelzl@29803
  1164
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
hoelzl@29803
  1165
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
hoelzl@29803
  1166
  shows "f x = f y"
hoelzl@29803
  1167
proof (cases "x = y")
hoelzl@29803
  1168
  case False
hoelzl@29803
  1169
  let ?a = "min x y"
hoelzl@29803
  1170
  let ?b = "max x y"
hoelzl@29803
  1171
  
hoelzl@29803
  1172
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
hoelzl@29803
  1173
  proof (rule allI, rule impI)
hoelzl@29803
  1174
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
hoelzl@29803
  1175
    hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
hoelzl@29803
  1176
    hence "z \<in> {a<..<b}" by auto
hoelzl@29803
  1177
    thus "DERIV f z :> 0" by (rule derivable)
hoelzl@29803
  1178
  qed
hoelzl@29803
  1179
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
hoelzl@29803
  1180
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
hoelzl@29803
  1181
hoelzl@29803
  1182
  have "?a < ?b" using `x \<noteq> y` by auto
hoelzl@29803
  1183
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
hoelzl@29803
  1184
  show ?thesis by auto
hoelzl@29803
  1185
qed auto
hoelzl@29803
  1186
huffman@21164
  1187
lemma DERIV_isconst_all:
huffman@21164
  1188
  fixes f :: "real => real"
huffman@21164
  1189
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1190
apply (rule linorder_cases [of x y])
huffman@21164
  1191
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1192
done
huffman@21164
  1193
huffman@21164
  1194
lemma DERIV_const_ratio_const:
huffman@21784
  1195
  fixes f :: "real => real"
huffman@21784
  1196
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1197
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1198
apply (drule_tac [!] f = f in MVT)
hoelzl@56181
  1199
apply (auto dest: DERIV_isCont DERIV_unique simp add: real_differentiable_def)
haftmann@54230
  1200
apply (auto dest: DERIV_unique simp add: ring_distribs)
huffman@21164
  1201
done
huffman@21164
  1202
huffman@21164
  1203
lemma DERIV_const_ratio_const2:
huffman@21784
  1204
  fixes f :: "real => real"
huffman@21784
  1205
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164
  1206
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164
  1207
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1208
done
huffman@21164
  1209
huffman@21164
  1210
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1211
by (simp)
huffman@21164
  1212
huffman@21164
  1213
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1214
by (simp)
huffman@21164
  1215
huffman@21164
  1216
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1217
huffman@21164
  1218
lemma DERIV_const_average:
huffman@21164
  1219
  fixes v :: "real => real"
huffman@21164
  1220
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1221
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1222
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1223
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1224
  case equal with neq show ?thesis by simp
huffman@21164
  1225
next
huffman@21164
  1226
  case less
huffman@21164
  1227
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1228
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1229
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1230
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1231
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1232
  ultimately show ?thesis using neq by force
huffman@21164
  1233
next
huffman@21164
  1234
  case greater
huffman@21164
  1235
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1236
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1237
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1238
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1239
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1240
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1241
qed
huffman@21164
  1242
paulson@33654
  1243
(* A function with positive derivative is increasing. 
paulson@33654
  1244
   A simple proof using the MVT, by Jeremy Avigad. And variants.
paulson@33654
  1245
*)
paulson@33654
  1246
lemma DERIV_pos_imp_increasing:
paulson@33654
  1247
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1248
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
paulson@33654
  1249
  shows "f a < f b"
paulson@33654
  1250
proof (rule ccontr)
wenzelm@41550
  1251
  assume f: "~ f a < f b"
wenzelm@33690
  1252
  have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1253
      & f b - f a = (b - a) * l"
wenzelm@33690
  1254
    apply (rule MVT)
wenzelm@33690
  1255
      using assms
wenzelm@33690
  1256
      apply auto
wenzelm@33690
  1257
      apply (metis DERIV_isCont)
huffman@36777
  1258
     apply (metis differentiableI less_le)
wenzelm@33690
  1259
    done
wenzelm@41550
  1260
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
paulson@33654
  1261
      and "f b - f a = (b - a) * l"
paulson@33654
  1262
    by auto
wenzelm@41550
  1263
  with assms f have "~(l > 0)"
huffman@36777
  1264
    by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
wenzelm@41550
  1265
  with assms z show False
huffman@36777
  1266
    by (metis DERIV_unique less_le)
paulson@33654
  1267
qed
paulson@33654
  1268
noschinl@45791
  1269
lemma DERIV_nonneg_imp_nondecreasing:
paulson@33654
  1270
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1271
  assumes "a \<le> b" and
paulson@33654
  1272
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
paulson@33654
  1273
  shows "f a \<le> f b"
paulson@33654
  1274
proof (rule ccontr, cases "a = b")
wenzelm@41550
  1275
  assume "~ f a \<le> f b" and "a = b"
wenzelm@41550
  1276
  then show False by auto
haftmann@37891
  1277
next
haftmann@37891
  1278
  assume A: "~ f a \<le> f b"
haftmann@37891
  1279
  assume B: "a ~= b"
paulson@33654
  1280
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1281
      & f b - f a = (b - a) * l"
wenzelm@33690
  1282
    apply -
wenzelm@33690
  1283
    apply (rule MVT)
wenzelm@33690
  1284
      apply auto
wenzelm@33690
  1285
      apply (metis DERIV_isCont)
huffman@36777
  1286
     apply (metis differentiableI less_le)
paulson@33654
  1287
    done
wenzelm@41550
  1288
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
haftmann@37891
  1289
      and C: "f b - f a = (b - a) * l"
paulson@33654
  1290
    by auto
haftmann@37891
  1291
  with A have "a < b" "f b < f a" by auto
haftmann@37891
  1292
  with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
huffman@45051
  1293
    (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
wenzelm@41550
  1294
  with assms z show False
paulson@33654
  1295
    by (metis DERIV_unique order_less_imp_le)
paulson@33654
  1296
qed
paulson@33654
  1297
paulson@33654
  1298
lemma DERIV_neg_imp_decreasing:
paulson@33654
  1299
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1300
  assumes "a < b" and
paulson@33654
  1301
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
paulson@33654
  1302
  shows "f a > f b"
paulson@33654
  1303
proof -
paulson@33654
  1304
  have "(%x. -f x) a < (%x. -f x) b"
paulson@33654
  1305
    apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
wenzelm@33690
  1306
    using assms
wenzelm@33690
  1307
    apply auto
paulson@33654
  1308
    apply (metis DERIV_minus neg_0_less_iff_less)
paulson@33654
  1309
    done
paulson@33654
  1310
  thus ?thesis
paulson@33654
  1311
    by simp
paulson@33654
  1312
qed
paulson@33654
  1313
paulson@33654
  1314
lemma DERIV_nonpos_imp_nonincreasing:
paulson@33654
  1315
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1316
  assumes "a \<le> b" and
paulson@33654
  1317
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
paulson@33654
  1318
  shows "f a \<ge> f b"
paulson@33654
  1319
proof -
paulson@33654
  1320
  have "(%x. -f x) a \<le> (%x. -f x) b"
noschinl@45791
  1321
    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
wenzelm@33690
  1322
    using assms
wenzelm@33690
  1323
    apply auto
paulson@33654
  1324
    apply (metis DERIV_minus neg_0_le_iff_le)
paulson@33654
  1325
    done
paulson@33654
  1326
  thus ?thesis
paulson@33654
  1327
    by simp
paulson@33654
  1328
qed
huffman@21164
  1329
huffman@23041
  1330
text {* Derivative of inverse function *}
huffman@23041
  1331
huffman@23041
  1332
lemma DERIV_inverse_function:
huffman@23041
  1333
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1334
  assumes der: "DERIV f (g x) :> D"
huffman@23041
  1335
  assumes neq: "D \<noteq> 0"
huffman@23044
  1336
  assumes a: "a < x" and b: "x < b"
huffman@23044
  1337
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041
  1338
  assumes cont: "isCont g x"
huffman@23041
  1339
  shows "DERIV g x :> inverse D"
huffman@23041
  1340
unfolding DERIV_iff2
huffman@23044
  1341
proof (rule LIM_equal2)
huffman@23044
  1342
  show "0 < min (x - a) (b - x)"
chaieb@27668
  1343
    using a b by arith 
huffman@23044
  1344
next
huffman@23041
  1345
  fix y
huffman@23044
  1346
  assume "norm (y - x) < min (x - a) (b - x)"
chaieb@27668
  1347
  hence "a < y" and "y < b" 
huffman@23044
  1348
    by (simp_all add: abs_less_iff)
huffman@23041
  1349
  thus "(g y - g x) / (y - x) =
huffman@23041
  1350
        inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1351
    by (simp add: inj)
huffman@23041
  1352
next
huffman@23041
  1353
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
huffman@23041
  1354
    by (rule der [unfolded DERIV_iff2])
huffman@23041
  1355
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
huffman@23044
  1356
    using inj a b by simp
huffman@23041
  1357
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
huffman@23041
  1358
  proof (safe intro!: exI)
huffman@23044
  1359
    show "0 < min (x - a) (b - x)"
huffman@23044
  1360
      using a b by simp
huffman@23041
  1361
  next
huffman@23041
  1362
    fix y
huffman@23044
  1363
    assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1364
    hence y: "a < y" "y < b"
huffman@23044
  1365
      by (simp_all add: abs_less_iff)
huffman@23041
  1366
    assume "g y = g x"
huffman@23041
  1367
    hence "f (g y) = f (g x)" by simp
huffman@23044
  1368
    hence "y = x" using inj y a b by simp
huffman@23041
  1369
    also assume "y \<noteq> x"
huffman@23041
  1370
    finally show False by simp
huffman@23041
  1371
  qed
huffman@23041
  1372
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
huffman@23041
  1373
    using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041
  1374
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
huffman@23041
  1375
        -- x --> inverse D"
huffman@44568
  1376
    using neq by (rule tendsto_inverse)
huffman@23041
  1377
qed
huffman@23041
  1378
huffman@29975
  1379
subsection {* Generalized Mean Value Theorem *}
huffman@29975
  1380
huffman@21164
  1381
theorem GMVT:
huffman@21784
  1382
  fixes a b :: real
huffman@21164
  1383
  assumes alb: "a < b"
wenzelm@41550
  1384
    and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
hoelzl@56181
  1385
    and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
wenzelm@41550
  1386
    and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
hoelzl@56181
  1387
    and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
wenzelm@53381
  1388
  shows "\<exists>g'c f'c c.
wenzelm@53381
  1389
    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)"
huffman@21164
  1390
proof -
huffman@21164
  1391
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
wenzelm@41550
  1392
  from assms have "a < b" by simp
huffman@21164
  1393
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@44233
  1394
    using fc gc by simp
hoelzl@56181
  1395
  moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
huffman@44233
  1396
    using fd gd by simp
huffman@21164
  1397
  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)
huffman@21164
  1398
  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" ..
huffman@21164
  1399
  then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
huffman@21164
  1400
huffman@21164
  1401
  from cdef have cint: "a < c \<and> c < b" by auto
hoelzl@56181
  1402
  with gd have "g differentiable (at c)" by simp
huffman@21164
  1403
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1404
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1405
huffman@21164
  1406
  from cdef have "a < c \<and> c < b" by auto
hoelzl@56181
  1407
  with fd have "f differentiable (at c)" by simp
huffman@21164
  1408
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1409
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1410
huffman@21164
  1411
  from cdef have "DERIV ?h c :> l" by auto
hoelzl@41368
  1412
  moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
hoelzl@41368
  1413
    using g'cdef f'cdef by (auto intro!: DERIV_intros)
huffman@21164
  1414
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1415
huffman@21164
  1416
  {
huffman@21164
  1417
    from cdef have "?h b - ?h a = (b - a) * l" by auto
wenzelm@53374
  1418
    also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1419
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1420
  }
huffman@21164
  1421
  moreover
huffman@21164
  1422
  {
huffman@21164
  1423
    have "?h b - ?h a =
huffman@21164
  1424
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1425
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
nipkow@29667
  1426
      by (simp add: algebra_simps)
huffman@21164
  1427
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1428
  }
huffman@21164
  1429
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1430
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1431
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1432
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1433
huffman@21164
  1434
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1435
qed
huffman@21164
  1436
hoelzl@50327
  1437
lemma GMVT':
hoelzl@50327
  1438
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1439
  assumes "a < b"
hoelzl@50327
  1440
  assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
hoelzl@50327
  1441
  assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
hoelzl@50327
  1442
  assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
hoelzl@50327
  1443
  assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
hoelzl@50327
  1444
  shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1445
proof -
hoelzl@50327
  1446
  have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
hoelzl@50327
  1447
    a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
hoelzl@56181
  1448
    using assms by (intro GMVT) (force simp: real_differentiable_def)+
hoelzl@50327
  1449
  then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1450
    using DERIV_f DERIV_g by (force dest: DERIV_unique)
hoelzl@50327
  1451
  then show ?thesis
hoelzl@50327
  1452
    by auto
hoelzl@50327
  1453
qed
hoelzl@50327
  1454
hoelzl@51529
  1455
hoelzl@51529
  1456
subsection {* L'Hopitals rule *}
hoelzl@51529
  1457
hoelzl@51641
  1458
lemma isCont_If_ge:
hoelzl@51641
  1459
  fixes a :: "'a :: linorder_topology"
hoelzl@51641
  1460
  shows "continuous (at_left a) g \<Longrightarrow> (f ---> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
hoelzl@51641
  1461
  unfolding isCont_def continuous_within
hoelzl@51641
  1462
  apply (intro filterlim_split_at)
hoelzl@51641
  1463
  apply (subst filterlim_cong[OF refl refl, where g=g])
hoelzl@51641
  1464
  apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1465
  apply (subst filterlim_cong[OF refl refl, where g=f])
hoelzl@51641
  1466
  apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1467
  done
hoelzl@51641
  1468
hoelzl@50327
  1469
lemma lhopital_right_0:
hoelzl@50329
  1470
  fixes f0 g0 :: "real \<Rightarrow> real"
hoelzl@50329
  1471
  assumes f_0: "(f0 ---> 0) (at_right 0)"
hoelzl@50329
  1472
  assumes g_0: "(g0 ---> 0) (at_right 0)"
hoelzl@50327
  1473
  assumes ev:
hoelzl@50329
  1474
    "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1475
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50329
  1476
    "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
hoelzl@50329
  1477
    "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
hoelzl@50327
  1478
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50329
  1479
  shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
hoelzl@50327
  1480
proof -
hoelzl@50329
  1481
  def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
hoelzl@50329
  1482
  then have "f 0 = 0" by simp
hoelzl@50329
  1483
hoelzl@50329
  1484
  def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
hoelzl@50329
  1485
  then have "g 0 = 0" by simp
hoelzl@50329
  1486
hoelzl@50329
  1487
  have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
hoelzl@50329
  1488
      DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
hoelzl@50329
  1489
    using ev by eventually_elim auto
hoelzl@50329
  1490
  then obtain a where [arith]: "0 < a"
hoelzl@50329
  1491
    and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
hoelzl@50327
  1492
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50329
  1493
    and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
hoelzl@50329
  1494
    and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
hoelzl@51641
  1495
    unfolding eventually_at eventually_at by (auto simp: dist_real_def)
hoelzl@50327
  1496
hoelzl@50329
  1497
  have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
hoelzl@50329
  1498
    using g0_neq_0 by (simp add: g_def)
hoelzl@50329
  1499
hoelzl@50329
  1500
  { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
hoelzl@50329
  1501
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
hoelzl@50329
  1502
         (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1503
  note f = this
hoelzl@50329
  1504
hoelzl@50329
  1505
  { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
hoelzl@50329
  1506
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
hoelzl@50329
  1507
         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1508
  note g = this
hoelzl@50329
  1509
hoelzl@50329
  1510
  have "isCont f 0"
hoelzl@51641
  1511
    unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
hoelzl@51641
  1512
hoelzl@50329
  1513
  have "isCont g 0"
hoelzl@51641
  1514
    unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
hoelzl@50329
  1515
hoelzl@50327
  1516
  have "\<exists>\<zeta>. \<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)"
hoelzl@50327
  1517
  proof (rule bchoice, rule)
hoelzl@50327
  1518
    fix x assume "x \<in> {0 <..< a}"
hoelzl@50327
  1519
    then have x[arith]: "0 < x" "x < a" by auto
hoelzl@50327
  1520
    with g'_neq_0 g_neq_0 `g 0 = 0` have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
hoelzl@50327
  1521
      by auto
hoelzl@50328
  1522
    have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
hoelzl@50328
  1523
      using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1524
    moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
hoelzl@50328
  1525
      using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1526
    ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
hoelzl@50328
  1527
      using f g `x < a` by (intro GMVT') auto
wenzelm@53374
  1528
    then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c"
wenzelm@53374
  1529
      by blast
hoelzl@50327
  1530
    moreover
wenzelm@53374
  1531
    from * g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
hoelzl@50327
  1532
      by (simp add: field_simps)
hoelzl@50327
  1533
    ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
hoelzl@50327
  1534
      using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
hoelzl@50327
  1535
  qed
wenzelm@53381
  1536
  then obtain \<zeta> where "\<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)" ..
hoelzl@50327
  1537
  then have \<zeta>: "eventually (\<lambda>x. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)) (at_right 0)"
hoelzl@51641
  1538
    unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
hoelzl@50327
  1539
  moreover
hoelzl@50327
  1540
  from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
hoelzl@50327
  1541
    by eventually_elim auto
hoelzl@50327
  1542
  then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
hoelzl@50327
  1543
    by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
hoelzl@51641
  1544
       (auto intro: tendsto_const tendsto_ident_at)
hoelzl@50327
  1545
  then have "(\<zeta> ---> 0) (at_right 0)"
hoelzl@50327
  1546
    by (rule tendsto_norm_zero_cancel)
hoelzl@50327
  1547
  with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
hoelzl@51641
  1548
    by (auto elim!: eventually_elim1 simp: filterlim_at)
hoelzl@50327
  1549
  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
hoelzl@50327
  1550
    by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
hoelzl@50329
  1551
  ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
hoelzl@50328
  1552
    by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
hoelzl@50328
  1553
       (auto elim: eventually_elim1)
hoelzl@50329
  1554
  also have "?P \<longleftrightarrow> ?thesis"
hoelzl@51641
  1555
    by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
hoelzl@50329
  1556
  finally show ?thesis .
hoelzl@50327
  1557
qed
hoelzl@50327
  1558
hoelzl@50330
  1559
lemma lhopital_right:
hoelzl@50330
  1560
  "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1561
    eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1562
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1563
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1564
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1565
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330
  1566
  ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330
  1567
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1568
  by (rule lhopital_right_0)
hoelzl@50330
  1569
hoelzl@50330
  1570
lemma lhopital_left:
hoelzl@50330
  1571
  "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1572
    eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1573
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1574
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1575
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1576
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330
  1577
  ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330
  1578
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@50330
  1579
  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1580
hoelzl@50330
  1581
lemma lhopital:
hoelzl@50330
  1582
  "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1583
    eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1584
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1585
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1586
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330
  1587
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330
  1588
  ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330
  1589
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1590
  by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
hoelzl@50330
  1591
hoelzl@50327
  1592
lemma lhopital_right_0_at_top:
hoelzl@50327
  1593
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1594
  assumes g_0: "LIM x at_right 0. g x :> at_top"
hoelzl@50327
  1595
  assumes ev:
hoelzl@50327
  1596
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1597
    "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
hoelzl@50327
  1598
    "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
hoelzl@50327
  1599
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50327
  1600
  shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
hoelzl@50327
  1601
  unfolding tendsto_iff
hoelzl@50327
  1602
proof safe
hoelzl@50327
  1603
  fix e :: real assume "0 < e"
hoelzl@50327
  1604
hoelzl@50327
  1605
  with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
hoelzl@50327
  1606
  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
hoelzl@50327
  1607
  from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
hoelzl@50327
  1608
  obtain a where [arith]: "0 < a"
hoelzl@50327
  1609
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50327
  1610
    and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
hoelzl@50327
  1611
    and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
hoelzl@50327
  1612
    and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
hoelzl@51641
  1613
    unfolding eventually_at_le by (auto simp: dist_real_def)
hoelzl@51641
  1614
    
hoelzl@50327
  1615
hoelzl@50327
  1616
  from Df have
hoelzl@50328
  1617
    "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
hoelzl@51641
  1618
    unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
hoelzl@50327
  1619
hoelzl@50327
  1620
  moreover
hoelzl@50328
  1621
  have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
hoelzl@50346
  1622
    using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
hoelzl@50327
  1623
hoelzl@50327
  1624
  moreover
hoelzl@50327
  1625
  have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
hoelzl@50327
  1626
    using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
hoelzl@50327
  1627
    by (rule filterlim_compose)
hoelzl@50327
  1628
  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
hoelzl@50327
  1629
    by (intro tendsto_intros)
hoelzl@50327
  1630
  then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
hoelzl@50327
  1631
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1632
  from this[unfolded tendsto_iff, rule_format, of 1]
hoelzl@50327
  1633
  have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
hoelzl@50327
  1634
    by (auto elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50327
  1635
hoelzl@50327
  1636
  moreover
hoelzl@50327
  1637
  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) ---> norm ((f a - x * g a) * 0)) (at_right 0)"
hoelzl@50327
  1638
    by (intro tendsto_intros)
hoelzl@50327
  1639
  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
hoelzl@50327
  1640
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1641
  from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
hoelzl@50327
  1642
  have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
hoelzl@50327
  1643
    by (auto simp: dist_real_def)
hoelzl@50327
  1644
hoelzl@50327
  1645
  ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
hoelzl@50327
  1646
  proof eventually_elim
hoelzl@50327
  1647
    fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
hoelzl@50327
  1648
    assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
hoelzl@50327
  1649
hoelzl@50327
  1650
    have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
hoelzl@50327
  1651
      using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
wenzelm@53381
  1652
    then obtain y where [arith]: "t < y" "y < a"
wenzelm@53381
  1653
      and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
wenzelm@53381
  1654
      by blast
wenzelm@53381
  1655
    from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
hoelzl@50327
  1656
      using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
hoelzl@50327
  1657
hoelzl@50327
  1658
    have *: "f t / g t - x = ((f t - f a) / (g t - g a) - x) * (1 - g a / g t) + (f a - x * g a) / g t"
hoelzl@50327
  1659
      by (simp add: field_simps)
hoelzl@50327
  1660
    have "norm (f t / g t - x) \<le>
hoelzl@50327
  1661
        norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
hoelzl@50327
  1662
      unfolding * by (rule norm_triangle_ineq)
hoelzl@50327
  1663
    also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
hoelzl@50327
  1664
      by (simp add: abs_mult D_eq dist_real_def)
hoelzl@50327
  1665
    also have "\<dots> < (e / 4) * 2 + e / 2"
hoelzl@50327
  1666
      using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
hoelzl@50327
  1667
    finally show "dist (f t / g t) x < e"
hoelzl@50327
  1668
      by (simp add: dist_real_def)
hoelzl@50327
  1669
  qed
hoelzl@50327
  1670
qed
hoelzl@50327
  1671
hoelzl@50330
  1672
lemma lhopital_right_at_top:
hoelzl@50330
  1673
  "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1674
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1675
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1676
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1677
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330
  1678
    ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330
  1679
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1680
  by (rule lhopital_right_0_at_top)
hoelzl@50330
  1681
hoelzl@50330
  1682
lemma lhopital_left_at_top:
hoelzl@50330
  1683
  "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1684
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1685
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1686
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1687
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330
  1688
    ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330
  1689
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@50330
  1690
  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1691
hoelzl@50330
  1692
lemma lhopital_at_top:
hoelzl@50330
  1693
  "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1694
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1695
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1696
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330
  1697
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330
  1698
    ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330
  1699
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1700
  by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
hoelzl@50330
  1701
hoelzl@50347
  1702
lemma lhospital_at_top_at_top:
hoelzl@50347
  1703
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50347
  1704
  assumes g_0: "LIM x at_top. g x :> at_top"
hoelzl@50347
  1705
  assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
hoelzl@50347
  1706
  assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
hoelzl@50347
  1707
  assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
hoelzl@50347
  1708
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
hoelzl@50347
  1709
  shows "((\<lambda> x. f x / g x) ---> x) at_top"
hoelzl@50347
  1710
  unfolding filterlim_at_top_to_right
hoelzl@50347
  1711
proof (rule lhopital_right_0_at_top)
hoelzl@50347
  1712
  let ?F = "\<lambda>x. f (inverse x)"
hoelzl@50347
  1713
  let ?G = "\<lambda>x. g (inverse x)"
hoelzl@50347
  1714
  let ?R = "at_right (0::real)"
hoelzl@50347
  1715
  let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
hoelzl@50347
  1716
hoelzl@50347
  1717
  show "LIM x ?R. ?G x :> at_top"
hoelzl@50347
  1718
    using g_0 unfolding filterlim_at_top_to_right .
hoelzl@50347
  1719
hoelzl@50347
  1720
  show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
hoelzl@50347
  1721
    unfolding eventually_at_right_to_top
hoelzl@50347
  1722
    using Dg eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1723
    apply eventually_elim
hoelzl@50347
  1724
    apply (rule DERIV_cong)
hoelzl@50347
  1725
    apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347
  1726
    apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1727
    done
hoelzl@50347
  1728
hoelzl@50347
  1729
  show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
hoelzl@50347
  1730
    unfolding eventually_at_right_to_top
hoelzl@50347
  1731
    using Df eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1732
    apply eventually_elim
hoelzl@50347
  1733
    apply (rule DERIV_cong)
hoelzl@50347
  1734
    apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347
  1735
    apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1736
    done
hoelzl@50347
  1737
hoelzl@50347
  1738
  show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
hoelzl@50347
  1739
    unfolding eventually_at_right_to_top
hoelzl@50347
  1740
    using g' eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1741
    by eventually_elim auto
hoelzl@50347
  1742
    
hoelzl@50347
  1743
  show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
hoelzl@50347
  1744
    unfolding filterlim_at_right_to_top
hoelzl@50347
  1745
    apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
hoelzl@50347
  1746
    using eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1747
    by eventually_elim simp
hoelzl@50347
  1748
qed
hoelzl@50347
  1749
huffman@21164
  1750
end