src/HOL/Deriv.thy
 author wenzelm Wed Apr 10 21:20:35 2013 +0200 (2013-04-10) changeset 51692 ecd34f863242 parent 51642 400ec5ae7f8f child 53374 a14d2a854c02 permissions -rw-r--r--
tuned pretty layout: avoid nested Pretty.string_of, which merely happens to work with Isabelle/jEdit since formatting is delegated to Scala side;
declare command "print_case_translations" where it is actually defined;
```     1 (*  Title       : Deriv.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4     Author      : Brian Huffman
```
```     5     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
```
```     6     GMVT by Benjamin Porter, 2005
```
```     7 *)
```
```     8
```
```     9 header{* Differentiation *}
```
```    10
```
```    11 theory Deriv
```
```    12 imports Limits
```
```    13 begin
```
```    14
```
```    15 definition
```
```    16   -- {* Frechet derivative: D is derivative of function f at x within s *}
```
```    17   has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow>  bool"
```
```    18   (infixl "(has'_derivative)" 12)
```
```    19 where
```
```    20   "(f has_derivative f') F \<longleftrightarrow>
```
```    21     (bounded_linear f' \<and>
```
```    22      ((\<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)"
```
```    23
```
```    24 lemma FDERIV_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
```
```    25   by simp
```
```    26
```
```    27 ML {*
```
```    28
```
```    29 structure FDERIV_Intros = Named_Thms
```
```    30 (
```
```    31   val name = @{binding FDERIV_intros}
```
```    32   val description = "introduction rules for FDERIV"
```
```    33 )
```
```    34
```
```    35 *}
```
```    36
```
```    37 setup {*
```
```    38   FDERIV_Intros.setup #>
```
```    39   Global_Theory.add_thms_dynamic (@{binding FDERIV_eq_intros},
```
```    40     map_filter (try (fn thm => @{thm FDERIV_eq_rhs} OF [thm])) o FDERIV_Intros.get o Context.proof_of);
```
```    41 *}
```
```    42
```
```    43 lemma FDERIV_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
```
```    44   by (simp add: has_derivative_def)
```
```    45
```
```    46 lemma FDERIV_ident[FDERIV_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
```
```    47   by (simp add: has_derivative_def tendsto_const)
```
```    48
```
```    49 lemma FDERIV_const[FDERIV_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"
```
```    50   by (simp add: has_derivative_def tendsto_const)
```
```    51
```
```    52 lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
```
```    53
```
```    54 lemma (in bounded_linear) FDERIV:
```
```    55   "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
```
```    56   using assms unfolding has_derivative_def
```
```    57   apply safe
```
```    58   apply (erule bounded_linear_compose [OF local.bounded_linear])
```
```    59   apply (drule local.tendsto)
```
```    60   apply (simp add: local.scaleR local.diff local.add local.zero)
```
```    61   done
```
```    62
```
```    63 lemmas FDERIV_scaleR_right [FDERIV_intros] =
```
```    64   bounded_linear.FDERIV [OF bounded_linear_scaleR_right]
```
```    65
```
```    66 lemmas FDERIV_scaleR_left [FDERIV_intros] =
```
```    67   bounded_linear.FDERIV [OF bounded_linear_scaleR_left]
```
```    68
```
```    69 lemmas FDERIV_mult_right [FDERIV_intros] =
```
```    70   bounded_linear.FDERIV [OF bounded_linear_mult_right]
```
```    71
```
```    72 lemmas FDERIV_mult_left [FDERIV_intros] =
```
```    73   bounded_linear.FDERIV [OF bounded_linear_mult_left]
```
```    74
```
```    75 lemma FDERIV_add[simp, FDERIV_intros]:
```
```    76   assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F"
```
```    77   shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
```
```    78   unfolding has_derivative_def
```
```    79 proof safe
```
```    80   let ?x = "Lim F (\<lambda>x. x)"
```
```    81   let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
```
```    82   have "((\<lambda>x. ?D f f' x + ?D g g' x) ---> (0 + 0)) F"
```
```    83     using f g by (intro tendsto_add) (auto simp: has_derivative_def)
```
```    84   then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"
```
```    85     by (simp add: field_simps scaleR_add_right scaleR_diff_right)
```
```    86 qed (blast intro: bounded_linear_add f g FDERIV_bounded_linear)
```
```    87
```
```    88 lemma FDERIV_setsum[simp, FDERIV_intros]:
```
```    89   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F"
```
```    90   shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
```
```    91 proof cases
```
```    92   assume "finite I" from this f show ?thesis
```
```    93     by induct (simp_all add: f)
```
```    94 qed simp
```
```    95
```
```    96 lemma FDERIV_minus[simp, FDERIV_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
```
```    97   using FDERIV_scaleR_right[of f f' F "-1"] by simp
```
```    98
```
```    99 lemma FDERIV_diff[simp, FDERIV_intros]:
```
```   100   "(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"
```
```   101   by (simp only: diff_minus FDERIV_add FDERIV_minus)
```
```   102
```
```   103 abbreviation
```
```   104   -- {* Frechet derivative: D is derivative of function f at x within s *}
```
```   105   FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
```
```   106   ("(FDERIV (_)/ (_)/ : (_)/ :> (_))" [1000, 1000, 1000, 60] 60)
```
```   107 where
```
```   108   "FDERIV f x : s :> f' \<equiv> (f has_derivative f') (at x within s)"
```
```   109
```
```   110 abbreviation
```
```   111   fderiv_at ::
```
```   112     "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
```
```   113     ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
```
```   114 where
```
```   115   "FDERIV f x :> D \<equiv> FDERIV f x : UNIV :> D"
```
```   116
```
```   117 lemma FDERIV_def:
```
```   118   "FDERIV f x : s :> f' \<longleftrightarrow>
```
```   119     (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s))"
```
```   120   by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)
```
```   121
```
```   122 lemma FDERIV_iff_norm:
```
```   123   "FDERIV f x : s :> f' \<longleftrightarrow>
```
```   124     (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ---> 0) (at x within s))"
```
```   125   using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
```
```   126   by (simp add: FDERIV_def divide_inverse ac_simps)
```
```   127
```
```   128 lemma fderiv_def:
```
```   129   "FDERIV f x :> D = (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0)"
```
```   130   unfolding FDERIV_iff_norm LIM_offset_zero_iff[of _ _ x] by simp
```
```   131
```
```   132 lemma field_fderiv_def:
```
```   133   fixes x :: "'a::real_normed_field"
```
```   134   shows "FDERIV f x :> (\<lambda>h. h * D) = (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
```
```   135   apply (unfold fderiv_def)
```
```   136   apply (simp add: bounded_linear_mult_left)
```
```   137   apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
```
```   138   apply (subst diff_divide_distrib)
```
```   139   apply (subst times_divide_eq_left [symmetric])
```
```   140   apply (simp cong: LIM_cong)
```
```   141   apply (simp add: tendsto_norm_zero_iff LIM_zero_iff)
```
```   142   done
```
```   143
```
```   144 lemma FDERIV_I:
```
```   145   "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s) \<Longrightarrow>
```
```   146   FDERIV f x : s :> f'"
```
```   147   by (simp add: FDERIV_def)
```
```   148
```
```   149 lemma FDERIV_I_sandwich:
```
```   150   assumes e: "0 < e" and bounded: "bounded_linear f'"
```
```   151     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)"
```
```   152     and "(H ---> 0) (at x within s)"
```
```   153   shows "FDERIV f x : s :> f'"
```
```   154   unfolding FDERIV_iff_norm
```
```   155 proof safe
```
```   156   show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) ---> 0) (at x within s)"
```
```   157   proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
```
```   158     show "(H ---> 0) (at x within s)" by fact
```
```   159     show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
```
```   160       unfolding eventually_at using e sandwich by auto
```
```   161   qed (auto simp: le_divide_eq tendsto_const)
```
```   162 qed fact
```
```   163
```
```   164 lemma FDERIV_subset: "FDERIV f x : s :> f' \<Longrightarrow> t \<subseteq> s \<Longrightarrow> FDERIV f x : t :> f'"
```
```   165   by (auto simp add: FDERIV_iff_norm intro: tendsto_within_subset)
```
```   166
```
```   167 subsection {* Continuity *}
```
```   168
```
```   169 lemma FDERIV_continuous:
```
```   170   assumes f: "FDERIV f x : s :> f'"
```
```   171   shows "continuous (at x within s) f"
```
```   172 proof -
```
```   173   from f interpret F: bounded_linear f' by (rule FDERIV_bounded_linear)
```
```   174   note F.tendsto[tendsto_intros]
```
```   175   let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
```
```   176   have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
```
```   177     using f unfolding FDERIV_iff_norm by blast
```
```   178   then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
```
```   179     by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
```
```   180   also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
```
```   181     by (intro filterlim_cong) (simp_all add: eventually_at_filter)
```
```   182   finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
```
```   183     by (rule tendsto_norm_zero_cancel)
```
```   184   then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
```
```   185     by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
```
```   186   then have "?L (\<lambda>y. f y - f x)"
```
```   187     by simp
```
```   188   from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
```
```   189     by (simp add: continuous_within)
```
```   190 qed
```
```   191
```
```   192 subsection {* Composition *}
```
```   193
```
```   194 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))"
```
```   195   unfolding tendsto_def eventually_inf_principal eventually_at_filter
```
```   196   by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
```
```   197
```
```   198 lemma FDERIV_in_compose:
```
```   199   assumes f: "FDERIV f x : s :> f'"
```
```   200   assumes g: "FDERIV g (f x) : (f`s) :> g'"
```
```   201   shows "FDERIV (\<lambda>x. g (f x)) x : s :> (\<lambda>x. g' (f' x))"
```
```   202 proof -
```
```   203   from f interpret F: bounded_linear f' by (rule FDERIV_bounded_linear)
```
```   204   from g interpret G: bounded_linear g' by (rule FDERIV_bounded_linear)
```
```   205   from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF" by fast
```
```   206   from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast
```
```   207   note G.tendsto[tendsto_intros]
```
```   208
```
```   209   let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
```
```   210   let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
```
```   211   let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
```
```   212   let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
```
```   213   def Nf \<equiv> "?N f f' x"
```
```   214   def Ng \<equiv> "\<lambda>y. ?N g g' (f x) (f y)"
```
```   215
```
```   216   show ?thesis
```
```   217   proof (rule FDERIV_I_sandwich[of 1])
```
```   218     show "bounded_linear (\<lambda>x. g' (f' x))"
```
```   219       using f g by (blast intro: bounded_linear_compose FDERIV_bounded_linear)
```
```   220   next
```
```   221     fix y::'a assume neq: "y \<noteq> x"
```
```   222     have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
```
```   223       by (simp add: G.diff G.add field_simps)
```
```   224     also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
```
```   225       by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def)
```
```   226     also have "\<dots> \<le> Nf y * kG + Ng y * (Nf y + kF)"
```
```   227     proof (intro add_mono mult_left_mono)
```
```   228       have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"
```
```   229         by simp
```
```   230       also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
```
```   231         by (rule norm_triangle_ineq)
```
```   232       also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
```
```   233         using kF by (intro add_mono) simp
```
```   234       finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
```
```   235         by (simp add: neq Nf_def field_simps)
```
```   236     qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
```
```   237     finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
```
```   238   next
```
```   239     have [tendsto_intros]: "?L Nf"
```
```   240       using f unfolding FDERIV_iff_norm Nf_def ..
```
```   241     from f have "(f ---> f x) (at x within s)"
```
```   242       by (blast intro: FDERIV_continuous continuous_within[THEN iffD1])
```
```   243     then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
```
```   244       unfolding filterlim_def
```
```   245       by (simp add: eventually_filtermap eventually_at_filter le_principal)
```
```   246
```
```   247     have "((?N g  g' (f x)) ---> 0) (at (f x) within f`s)"
```
```   248       using g unfolding FDERIV_iff_norm ..
```
```   249     then have g': "((?N g  g' (f x)) ---> 0) (inf (nhds (f x)) (principal (f`s)))"
```
```   250       by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
```
```   251
```
```   252     have [tendsto_intros]: "?L Ng"
```
```   253       unfolding Ng_def by (rule filterlim_compose[OF g' f'])
```
```   254     show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) ---> 0) (at x within s)"
```
```   255       by (intro tendsto_eq_intros) auto
```
```   256   qed simp
```
```   257 qed
```
```   258
```
```   259 lemma FDERIV_compose:
```
```   260   "FDERIV f x : s :> f' \<Longrightarrow> FDERIV g (f x) :> g' \<Longrightarrow> FDERIV (\<lambda>x. g (f x)) x : s :> (\<lambda>x. g' (f' x))"
```
```   261   by (blast intro: FDERIV_in_compose FDERIV_subset)
```
```   262
```
```   263 lemma (in bounded_bilinear) FDERIV:
```
```   264   assumes f: "FDERIV f x : s :> f'" and g: "FDERIV g x : s :> g'"
```
```   265   shows "FDERIV (\<lambda>x. f x ** g x) x : s :> (\<lambda>h. f x ** g' h + f' h ** g x)"
```
```   266 proof -
```
```   267   from bounded_linear.bounded [OF FDERIV_bounded_linear [OF f]]
```
```   268   obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
```
```   269
```
```   270   from pos_bounded obtain K where K: "0 < K" and norm_prod:
```
```   271     "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
```
```   272   let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
```
```   273   let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
```
```   274   def Ng =="?N g g'" and Nf =="?N f f'"
```
```   275
```
```   276   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)"
```
```   277   let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
```
```   278   let ?F = "at x within s"
```
```   279
```
```   280   show ?thesis
```
```   281   proof (rule FDERIV_I_sandwich[of 1])
```
```   282     show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
```
```   283       by (intro bounded_linear_add
```
```   284         bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
```
```   285         FDERIV_bounded_linear [OF g] FDERIV_bounded_linear [OF f])
```
```   286   next
```
```   287     from g have "(g ---> g x) ?F"
```
```   288       by (intro continuous_within[THEN iffD1] FDERIV_continuous)
```
```   289     moreover from f g have "(Nf ---> 0) ?F" "(Ng ---> 0) ?F"
```
```   290       by (simp_all add: FDERIV_iff_norm Ng_def Nf_def)
```
```   291     ultimately have "(?fun2 ---> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
```
```   292       by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
```
```   293     then show "(?fun2 ---> 0) ?F"
```
```   294       by simp
```
```   295   next
```
```   296     fix y::'d assume "y \<noteq> x"
```
```   297     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)"
```
```   298       by (simp add: diff_left diff_right add_left add_right field_simps)
```
```   299     also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
```
```   300         norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
```
```   301       by (intro divide_right_mono mult_mono'
```
```   302                 order_trans [OF norm_triangle_ineq add_mono]
```
```   303                 order_trans [OF norm_prod mult_right_mono]
```
```   304                 mult_nonneg_nonneg order_refl norm_ge_zero norm_F
```
```   305                 K [THEN order_less_imp_le])
```
```   306     also have "\<dots> = ?fun2 y"
```
```   307       by (simp add: add_divide_distrib Ng_def Nf_def)
```
```   308     finally show "?fun1 y \<le> ?fun2 y" .
```
```   309   qed simp
```
```   310 qed
```
```   311
```
```   312 lemmas FDERIV_mult[simp, FDERIV_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
```
```   313 lemmas FDERIV_scaleR[simp, FDERIV_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
```
```   314
```
```   315 lemma FDERIV_setprod[simp, FDERIV_intros]:
```
```   316   fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
```
```   317   assumes f: "\<And>i. i \<in> I \<Longrightarrow> FDERIV (f i) x : s :> f' i"
```
```   318   shows "FDERIV (\<lambda>x. \<Prod>i\<in>I. f i x) x : s :> (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))"
```
```   319 proof cases
```
```   320   assume "finite I" from this f show ?thesis
```
```   321   proof induct
```
```   322     case (insert i I)
```
```   323     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)"
```
```   324     have "FDERIV (\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) x : s :> ?P"
```
```   325       using insert by (intro FDERIV_mult) auto
```
```   326     also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
```
```   327       using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum_cong)
```
```   328     finally show ?case
```
```   329       using insert by simp
```
```   330   qed simp
```
```   331 qed simp
```
```   332
```
```   333 lemma FDERIV_power[simp, FDERIV_intros]:
```
```   334   fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
```
```   335   assumes f: "FDERIV f x : s :> f'"
```
```   336   shows "FDERIV  (\<lambda>x. f x^n) x : s :> (\<lambda>y. of_nat n * f' y * f x^(n - 1))"
```
```   337   using FDERIV_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)
```
```   338
```
```   339 lemma FDERIV_inverse':
```
```   340   fixes x :: "'a::real_normed_div_algebra"
```
```   341   assumes x: "x \<noteq> 0"
```
```   342   shows "FDERIV inverse x : s :> (\<lambda>h. - (inverse x * h * inverse x))"
```
```   343         (is "FDERIV ?inv x : s :> ?f")
```
```   344 proof (rule FDERIV_I_sandwich)
```
```   345   show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
```
```   346     apply (rule bounded_linear_minus)
```
```   347     apply (rule bounded_linear_mult_const)
```
```   348     apply (rule bounded_linear_const_mult)
```
```   349     apply (rule bounded_linear_ident)
```
```   350     done
```
```   351 next
```
```   352   show "0 < norm x" using x by simp
```
```   353 next
```
```   354   show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) ---> 0) (at x within s)"
```
```   355     apply (rule tendsto_mult_left_zero)
```
```   356     apply (rule tendsto_norm_zero)
```
```   357     apply (rule LIM_zero)
```
```   358     apply (rule tendsto_inverse)
```
```   359     apply (rule tendsto_ident_at)
```
```   360     apply (rule x)
```
```   361     done
```
```   362 next
```
```   363   fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"
```
```   364   then have "y \<noteq> 0"
```
```   365     by (auto simp: norm_conv_dist dist_commute)
```
```   366   have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
```
```   367     apply (subst inverse_diff_inverse [OF `y \<noteq> 0` x])
```
```   368     apply (subst minus_diff_minus)
```
```   369     apply (subst norm_minus_cancel)
```
```   370     apply (simp add: left_diff_distrib)
```
```   371     done
```
```   372   also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
```
```   373     apply (rule divide_right_mono [OF _ norm_ge_zero])
```
```   374     apply (rule order_trans [OF norm_mult_ineq])
```
```   375     apply (rule mult_right_mono [OF _ norm_ge_zero])
```
```   376     apply (rule norm_mult_ineq)
```
```   377     done
```
```   378   also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
```
```   379     by simp
```
```   380   finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
```
```   381       norm (?inv y - ?inv x) * norm (?inv x)" .
```
```   382 qed
```
```   383
```
```   384 lemma FDERIV_inverse[simp, FDERIV_intros]:
```
```   385   fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
```
```   386   assumes x:  "f x \<noteq> 0" and f: "FDERIV f x : s :> f'"
```
```   387   shows "FDERIV (\<lambda>x. inverse (f x)) x : s :> (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))"
```
```   388   using FDERIV_compose[OF f FDERIV_inverse', OF x] .
```
```   389
```
```   390 lemma FDERIV_divide[simp, FDERIV_intros]:
```
```   391   fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
```
```   392   assumes g: "FDERIV g x : s :> g'"
```
```   393   assumes x:  "f x \<noteq> 0" and f: "FDERIV f x : s :> f'"
```
```   394   shows "FDERIV (\<lambda>x. g x / f x) x : s :> (\<lambda>h. - g x * (inverse (f x) * f' h * inverse (f x)) + g' h / f x)"
```
```   395   using FDERIV_mult[OF g FDERIV_inverse[OF x f]]
```
```   396   by (simp add: divide_inverse)
```
```   397
```
```   398 subsection {* Uniqueness *}
```
```   399
```
```   400 text {*
```
```   401
```
```   402 This can not generally shown for @{const FDERIV}, as we need to approach the point from
```
```   403 all directions. There is a proof in @{text Multivariate_Analysis} for @{text euclidean_space}.
```
```   404
```
```   405 *}
```
```   406
```
```   407 lemma FDERIV_zero_unique:
```
```   408   assumes "FDERIV (\<lambda>x. 0) x :> F" shows "F = (\<lambda>h. 0)"
```
```   409 proof -
```
```   410   interpret F: bounded_linear F
```
```   411     using assms by (rule FDERIV_bounded_linear)
```
```   412   let ?r = "\<lambda>h. norm (F h) / norm h"
```
```   413   have *: "?r -- 0 --> 0"
```
```   414     using assms unfolding fderiv_def by simp
```
```   415   show "F = (\<lambda>h. 0)"
```
```   416   proof
```
```   417     fix h show "F h = 0"
```
```   418     proof (rule ccontr)
```
```   419       assume "F h \<noteq> 0"
```
```   420       moreover from this have h: "h \<noteq> 0"
```
```   421         by (clarsimp simp add: F.zero)
```
```   422       ultimately have "0 < ?r h"
```
```   423         by (simp add: divide_pos_pos)
```
```   424       from LIM_D [OF * this] obtain s where s: "0 < s"
```
```   425         and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
```
```   426       from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
```
```   427       let ?x = "scaleR (t / norm h) h"
```
```   428       have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
```
```   429       hence "?r ?x < ?r h" by (rule r)
```
```   430       thus "False" using t h by (simp add: F.scaleR)
```
```   431     qed
```
```   432   qed
```
```   433 qed
```
```   434
```
```   435 lemma FDERIV_unique:
```
```   436   assumes "FDERIV f x :> F" and "FDERIV f x :> F'" shows "F = F'"
```
```   437 proof -
```
```   438   have "FDERIV (\<lambda>x. 0) x :> (\<lambda>h. F h - F' h)"
```
```   439     using FDERIV_diff [OF assms] by simp
```
```   440   hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
```
```   441     by (rule FDERIV_zero_unique)
```
```   442   thus "F = F'"
```
```   443     unfolding fun_eq_iff right_minus_eq .
```
```   444 qed
```
```   445
```
```   446 subsection {* Differentiability predicate *}
```
```   447
```
```   448 definition isDiff :: "'a filter \<Rightarrow> ('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> bool" where
```
```   449   isDiff_def: "isDiff F f \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
```
```   450
```
```   451 abbreviation differentiable_in :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool"
```
```   452     ("(_) differentiable (_) in (_)"  [1000, 1000, 60] 60) where
```
```   453   "f differentiable x in s \<equiv> isDiff (at x within s) f"
```
```   454
```
```   455 abbreviation differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool"
```
```   456     (infixl "differentiable" 60) where
```
```   457   "f differentiable x \<equiv> f differentiable x in UNIV"
```
```   458
```
```   459 lemma differentiable_subset: "f differentiable x in s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable x in t"
```
```   460   unfolding isDiff_def by (blast intro: FDERIV_subset)
```
```   461
```
```   462 lemma differentiable_ident [simp]: "isDiff F (\<lambda>x. x)"
```
```   463   unfolding isDiff_def by (blast intro: FDERIV_ident)
```
```   464
```
```   465 lemma differentiable_const [simp]: "isDiff F (\<lambda>z. a)"
```
```   466   unfolding isDiff_def by (blast intro: FDERIV_const)
```
```   467
```
```   468 lemma differentiable_in_compose:
```
```   469   "f differentiable (g x) in (g`s) \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f (g x)) differentiable x in s"
```
```   470   unfolding isDiff_def by (blast intro: FDERIV_in_compose)
```
```   471
```
```   472 lemma differentiable_compose:
```
```   473   "f differentiable (g x) \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f (g x)) differentiable x in s"
```
```   474   by (blast intro: differentiable_in_compose differentiable_subset)
```
```   475
```
```   476 lemma differentiable_sum [simp]:
```
```   477   "isDiff F f \<Longrightarrow> isDiff F g \<Longrightarrow> isDiff F (\<lambda>x. f x + g x)"
```
```   478   unfolding isDiff_def by (blast intro: FDERIV_add)
```
```   479
```
```   480 lemma differentiable_minus [simp]:
```
```   481   "isDiff F f \<Longrightarrow> isDiff F (\<lambda>x. - f x)"
```
```   482   unfolding isDiff_def by (blast intro: FDERIV_minus)
```
```   483
```
```   484 lemma differentiable_diff [simp]:
```
```   485   "isDiff F f \<Longrightarrow> isDiff F g \<Longrightarrow> isDiff F (\<lambda>x. f x - g x)"
```
```   486   unfolding isDiff_def by (blast intro: FDERIV_diff)
```
```   487
```
```   488 lemma differentiable_mult [simp]:
```
```   489   fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
```
```   490   shows "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f x * g x) differentiable x in s"
```
```   491   unfolding isDiff_def by (blast intro: FDERIV_mult)
```
```   492
```
```   493 lemma differentiable_inverse [simp]:
```
```   494   fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
```
```   495   shows "f differentiable x in s \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable x in s"
```
```   496   unfolding isDiff_def by (blast intro: FDERIV_inverse)
```
```   497
```
```   498 lemma differentiable_divide [simp]:
```
```   499   fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
```
```   500   shows "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable x in s"
```
```   501   unfolding divide_inverse using assms by simp
```
```   502
```
```   503 lemma differentiable_power [simp]:
```
```   504   fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
```
```   505   shows "f differentiable x in s \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable x in s"
```
```   506   unfolding isDiff_def by (blast intro: FDERIV_power)
```
```   507
```
```   508 lemma differentiable_scaleR [simp]:
```
```   509   "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable x in s"
```
```   510   unfolding isDiff_def by (blast intro: FDERIV_scaleR)
```
```   511
```
```   512 definition
```
```   513   -- {*Differentiation: D is derivative of function f at x*}
```
```   514   deriv ::
```
```   515     "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool"
```
```   516     ("(DERIV (_)/ (_)/ : (_)/ :> (_))" [1000, 1000, 1000, 60] 60)
```
```   517 where
```
```   518   deriv_fderiv: "DERIV f x : s :> D = FDERIV f x : s :> (\<lambda>x. x * D)"
```
```   519
```
```   520 abbreviation
```
```   521   -- {*Differentiation: D is derivative of function f at x*}
```
```   522   deriv_at ::
```
```   523     "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   524     ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
```
```   525 where
```
```   526   "DERIV f x :> D \<equiv> DERIV f x : UNIV :> D"
```
```   527
```
```   528 lemma differentiable_def: "(f::real \<Rightarrow> real) differentiable x in s \<longleftrightarrow> (\<exists>D. DERIV f x : s :> D)"
```
```   529 proof safe
```
```   530   assume "f differentiable x in s"
```
```   531   then obtain f' where "FDERIV f x : s :> f'"
```
```   532     unfolding isDiff_def by auto
```
```   533   moreover then obtain c where "f' = (\<lambda>x. x * c)"
```
```   534     by (metis real_bounded_linear FDERIV_bounded_linear)
```
```   535   ultimately show "\<exists>D. DERIV f x : s :> D"
```
```   536     unfolding deriv_fderiv by auto
```
```   537 qed (auto simp: isDiff_def deriv_fderiv)
```
```   538
```
```   539 lemma differentiableE [elim?]:
```
```   540   fixes f :: "real \<Rightarrow> real"
```
```   541   assumes f: "f differentiable x in s" obtains df where "DERIV f x : s :> df"
```
```   542   using assms by (auto simp: differentiable_def)
```
```   543
```
```   544 lemma differentiableD: "(f::real \<Rightarrow> real) differentiable x in s \<Longrightarrow> \<exists>D. DERIV f x : s :> D"
```
```   545   by (auto elim: differentiableE)
```
```   546
```
```   547 lemma differentiableI: "DERIV f x : s :> D \<Longrightarrow> (f::real \<Rightarrow> real) differentiable x in s"
```
```   548   by (force simp add: differentiable_def)
```
```   549
```
```   550 lemma DERIV_I_FDERIV: "FDERIV f x : s :> F \<Longrightarrow> (\<And>x. x * F' = F x) \<Longrightarrow> DERIV f x : s :> F'"
```
```   551   by (simp add: deriv_fderiv)
```
```   552
```
```   553 lemma DERIV_D_FDERIV: "DERIV f x : s :> F \<Longrightarrow> FDERIV f x : s :> (\<lambda>x. x * F)"
```
```   554   by (simp add: deriv_fderiv)
```
```   555
```
```   556 lemma deriv_def:
```
```   557   "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
```
```   558   apply (simp add: deriv_fderiv fderiv_def bounded_linear_mult_left LIM_zero_iff[symmetric, of _ D])
```
```   559   apply (subst (2) tendsto_norm_zero_iff[symmetric])
```
```   560   apply (rule filterlim_cong)
```
```   561   apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
```
```   562   done
```
```   563
```
```   564 subsection {* Derivatives *}
```
```   565
```
```   566 lemma DERIV_iff: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
```
```   567   by (simp add: deriv_def)
```
```   568
```
```   569 lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
```
```   570   by (simp add: deriv_def)
```
```   571
```
```   572 lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x : s :> 0"
```
```   573   by (rule DERIV_I_FDERIV[OF FDERIV_const]) auto
```
```   574
```
```   575 lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x : s :> 1"
```
```   576   by (rule DERIV_I_FDERIV[OF FDERIV_ident]) auto
```
```   577
```
```   578 lemma DERIV_add: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x : s :> D + E"
```
```   579   by (rule DERIV_I_FDERIV[OF FDERIV_add]) (auto simp: field_simps dest: DERIV_D_FDERIV)
```
```   580
```
```   581 lemma DERIV_minus: "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x : s :> - D"
```
```   582   by (rule DERIV_I_FDERIV[OF FDERIV_minus]) (auto simp: field_simps dest: DERIV_D_FDERIV)
```
```   583
```
```   584 lemma DERIV_diff: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x : s :> D - E"
```
```   585   by (rule DERIV_I_FDERIV[OF FDERIV_diff]) (auto simp: field_simps dest: DERIV_D_FDERIV)
```
```   586
```
```   587 lemma DERIV_add_minus: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x : s :> D + - E"
```
```   588   by (simp only: DERIV_add DERIV_minus)
```
```   589
```
```   590 lemma DERIV_continuous: "DERIV f x : s :> D \<Longrightarrow> continuous (at x within s) f"
```
```   591   by (drule FDERIV_continuous[OF DERIV_D_FDERIV]) simp
```
```   592
```
```   593 lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
```
```   594   by (auto dest!: DERIV_continuous)
```
```   595
```
```   596 lemma DERIV_mult': "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x : s :> f x * E + D * g x"
```
```   597   by (rule DERIV_I_FDERIV[OF FDERIV_mult]) (auto simp: field_simps dest: DERIV_D_FDERIV)
```
```   598
```
```   599 lemma DERIV_mult: "DERIV f x : s :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x : s :> Da * g x + Db * f x"
```
```   600   by (rule DERIV_I_FDERIV[OF FDERIV_mult]) (auto simp: field_simps dest: DERIV_D_FDERIV)
```
```   601
```
```   602 text {* Derivative of linear multiplication *}
```
```   603
```
```   604 lemma DERIV_cmult:
```
```   605   "DERIV f x : s :> D ==> DERIV (%x. c * f x) x : s :> c*D"
```
```   606   by (drule DERIV_mult' [OF DERIV_const], simp)
```
```   607
```
```   608 lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x : s :> c"
```
```   609   by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
```
```   610
```
```   611 lemma DERIV_cdivide: "DERIV f x : s :> D ==> DERIV (%x. f x / c) x : s :> D / c"
```
```   612   apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x : s :> (1 / c) * D", force)
```
```   613   apply (erule DERIV_cmult)
```
```   614   done
```
```   615
```
```   616 lemma DERIV_unique:
```
```   617   "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
```
```   618   unfolding deriv_def by (rule LIM_unique)
```
```   619
```
```   620 lemma DERIV_setsum':
```
```   621   "(\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x : s :> (f' x n)) \<Longrightarrow> DERIV (\<lambda>x. setsum (f x) S) x : s :> setsum (f' x) S"
```
```   622   by (rule DERIV_I_FDERIV[OF FDERIV_setsum]) (auto simp: setsum_right_distrib dest: DERIV_D_FDERIV)
```
```   623
```
```   624 lemma DERIV_setsum:
```
```   625   "finite S \<Longrightarrow> (\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x : s :> (f' x n)) \<Longrightarrow> DERIV (\<lambda>x. setsum (f x) S) x : s :> setsum (f' x) S"
```
```   626   by (rule DERIV_setsum')
```
```   627
```
```   628 lemma DERIV_sumr [rule_format (no_asm)]: (* REMOVE *)
```
```   629      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x : s :> (f' r x))
```
```   630       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x : s :> (\<Sum>r=m..<n. f' r x)"
```
```   631   by (auto intro: DERIV_setsum)
```
```   632
```
```   633 lemma DERIV_inverse':
```
```   634   "DERIV f x : s :> D \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse (f x)) x : s :> - (inverse (f x) * D * inverse (f x))"
```
```   635   by (rule DERIV_I_FDERIV[OF FDERIV_inverse]) (auto dest: DERIV_D_FDERIV)
```
```   636
```
```   637 text {* Power of @{text "-1"} *}
```
```   638
```
```   639 lemma DERIV_inverse:
```
```   640   "x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse(x)) x : s :> - (inverse x ^ Suc (Suc 0))"
```
```   641   by (drule DERIV_inverse' [OF DERIV_ident]) simp
```
```   642
```
```   643 text {* Derivative of inverse *}
```
```   644
```
```   645 lemma DERIV_inverse_fun:
```
```   646   "DERIV f x : s :> d \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse (f x)) x : s :> (- (d * inverse(f x ^ Suc (Suc 0))))"
```
```   647   by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
```
```   648
```
```   649 text {* Derivative of quotient *}
```
```   650
```
```   651 lemma DERIV_divide:
```
```   652   "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x : s :> (D * g x - f x * E) / (g x * g x)"
```
```   653   by (rule DERIV_I_FDERIV[OF FDERIV_divide])
```
```   654      (auto dest: DERIV_D_FDERIV simp: field_simps nonzero_inverse_mult_distrib divide_inverse)
```
```   655
```
```   656 lemma DERIV_quotient:
```
```   657   "DERIV f x : s :> d \<Longrightarrow> DERIV g x : s :> e \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>y. f y / g y) x : s :> (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))"
```
```   658   by (drule (2) DERIV_divide) (simp add: mult_commute)
```
```   659
```
```   660 lemma DERIV_power_Suc:
```
```   661   "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. f x ^ Suc n) x : s :> (1 + of_nat n) * (D * f x ^ n)"
```
```   662   by (rule DERIV_I_FDERIV[OF FDERIV_power]) (auto simp: deriv_fderiv)
```
```   663
```
```   664 lemma DERIV_power:
```
```   665   "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. f x ^ n) x : s :> of_nat n * (D * f x ^ (n - Suc 0))"
```
```   666   by (rule DERIV_I_FDERIV[OF FDERIV_power]) (auto simp: deriv_fderiv)
```
```   667
```
```   668 lemma DERIV_pow: "DERIV (%x. x ^ n) x : s :> real n * (x ^ (n - Suc 0))"
```
```   669   apply (cut_tac DERIV_power [OF DERIV_ident])
```
```   670   apply (simp add: real_of_nat_def)
```
```   671   done
```
```   672
```
```   673 lemma DERIV_chain': "DERIV f x : s :> D \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> DERIV (\<lambda>x. g (f x)) x : s :> E * D"
```
```   674   using FDERIV_compose[of f "\<lambda>x. x * D" x s g "\<lambda>x. x * E"]
```
```   675   by (auto simp: deriv_fderiv ac_simps dest: FDERIV_subset)
```
```   676
```
```   677 text {* Standard version *}
```
```   678
```
```   679 lemma DERIV_chain: "DERIV f (g x) :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (f o g) x : s :> Da * Db"
```
```   680   by (drule (1) DERIV_chain', simp add: o_def mult_commute)
```
```   681
```
```   682 lemma DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (%x. f (g x)) x : s :> Da * Db"
```
```   683   by (auto dest: DERIV_chain simp add: o_def)
```
```   684
```
```   685 subsubsection {* @{text "DERIV_intros"} *}
```
```   686
```
```   687 ML {*
```
```   688 structure Deriv_Intros = Named_Thms
```
```   689 (
```
```   690   val name = @{binding DERIV_intros}
```
```   691   val description = "DERIV introduction rules"
```
```   692 )
```
```   693 *}
```
```   694
```
```   695 setup Deriv_Intros.setup
```
```   696
```
```   697 lemma DERIV_cong: "DERIV f x : s :> X \<Longrightarrow> X = Y \<Longrightarrow> DERIV f x : s :> Y"
```
```   698   by simp
```
```   699
```
```   700 declare
```
```   701   DERIV_const[THEN DERIV_cong, DERIV_intros]
```
```   702   DERIV_ident[THEN DERIV_cong, DERIV_intros]
```
```   703   DERIV_add[THEN DERIV_cong, DERIV_intros]
```
```   704   DERIV_minus[THEN DERIV_cong, DERIV_intros]
```
```   705   DERIV_mult[THEN DERIV_cong, DERIV_intros]
```
```   706   DERIV_diff[THEN DERIV_cong, DERIV_intros]
```
```   707   DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
```
```   708   DERIV_divide[THEN DERIV_cong, DERIV_intros]
```
```   709   DERIV_power[where 'a=real, THEN DERIV_cong,
```
```   710               unfolded real_of_nat_def[symmetric], DERIV_intros]
```
```   711   DERIV_setsum[THEN DERIV_cong, DERIV_intros]
```
```   712
```
```   713 text{*Alternative definition for differentiability*}
```
```   714
```
```   715 lemma DERIV_LIM_iff:
```
```   716   fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
```
```   717      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
```
```   718       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
```
```   719 apply (rule iffI)
```
```   720 apply (drule_tac k="- a" in LIM_offset)
```
```   721 apply (simp add: diff_minus)
```
```   722 apply (drule_tac k="a" in LIM_offset)
```
```   723 apply (simp add: add_commute)
```
```   724 done
```
```   725
```
```   726 lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"
```
```   727   by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
```
```   728
```
```   729 lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
```
```   730     DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
```
```   731   unfolding DERIV_iff2
```
```   732 proof (rule filterlim_cong)
```
```   733   assume "eventually (\<lambda>x. f x = g x) (nhds x)"
```
```   734   moreover then have "f x = g x" by (auto simp: eventually_nhds)
```
```   735   moreover assume "x = y" "u = v"
```
```   736   ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
```
```   737     by (auto simp: eventually_at_filter elim: eventually_elim1)
```
```   738 qed simp_all
```
```   739
```
```   740 lemma DERIV_shift:
```
```   741   "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
```
```   742   by (simp add: DERIV_iff field_simps)
```
```   743
```
```   744 lemma DERIV_mirror:
```
```   745   "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
```
```   746   by (simp add: deriv_def filterlim_at_split filterlim_at_left_to_right
```
```   747                 tendsto_minus_cancel_left field_simps conj_commute)
```
```   748
```
```   749 text {* Caratheodory formulation of derivative at a point *}
```
```   750
```
```   751 lemma CARAT_DERIV:
```
```   752   "(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)"
```
```   753       (is "?lhs = ?rhs")
```
```   754 proof
```
```   755   assume der: "DERIV f x :> l"
```
```   756   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
```
```   757   proof (intro exI conjI)
```
```   758     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
```
```   759     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
```
```   760     show "isCont ?g x" using der
```
```   761       by (simp add: isCont_iff DERIV_iff diff_minus
```
```   762                cong: LIM_equal [rule_format])
```
```   763     show "?g x = l" by simp
```
```   764   qed
```
```   765 next
```
```   766   assume "?rhs"
```
```   767   then obtain g where
```
```   768     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
```
```   769   thus "(DERIV f x :> l)"
```
```   770      by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
```
```   771 qed
```
```   772
```
```   773 text {*
```
```   774  Let's do the standard proof, though theorem
```
```   775  @{text "LIM_mult2"} follows from a NS proof
```
```   776 *}
```
```   777
```
```   778 subsection {* Local extrema *}
```
```   779
```
```   780 text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
```
```   781
```
```   782 lemma DERIV_pos_inc_right:
```
```   783   fixes f :: "real => real"
```
```   784   assumes der: "DERIV f x :> l"
```
```   785       and l:   "0 < l"
```
```   786   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
```
```   787 proof -
```
```   788   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
```
```   789   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)"
```
```   790     by (simp add: diff_minus)
```
```   791   then obtain s
```
```   792         where s:   "0 < s"
```
```   793           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
```
```   794     by auto
```
```   795   thus ?thesis
```
```   796   proof (intro exI conjI strip)
```
```   797     show "0<s" using s .
```
```   798     fix h::real
```
```   799     assume "0 < h" "h < s"
```
```   800     with all [of h] show "f x < f (x+h)"
```
```   801     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   802     split add: split_if_asm)
```
```   803       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
```
```   804       with l
```
```   805       have "0 < (f (x+h) - f x) / h" by arith
```
```   806       thus "f x < f (x+h)"
```
```   807   by (simp add: pos_less_divide_eq h)
```
```   808     qed
```
```   809   qed
```
```   810 qed
```
```   811
```
```   812 lemma DERIV_neg_dec_left:
```
```   813   fixes f :: "real => real"
```
```   814   assumes der: "DERIV f x :> l"
```
```   815       and l:   "l < 0"
```
```   816   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
```
```   817 proof -
```
```   818   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
```
```   819   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)"
```
```   820     by (simp add: diff_minus)
```
```   821   then obtain s
```
```   822         where s:   "0 < s"
```
```   823           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
```
```   824     by auto
```
```   825   thus ?thesis
```
```   826   proof (intro exI conjI strip)
```
```   827     show "0<s" using s .
```
```   828     fix h::real
```
```   829     assume "0 < h" "h < s"
```
```   830     with all [of "-h"] show "f x < f (x-h)"
```
```   831     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   832     split add: split_if_asm)
```
```   833       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
```
```   834       with l
```
```   835       have "0 < (f (x-h) - f x) / h" by arith
```
```   836       thus "f x < f (x-h)"
```
```   837   by (simp add: pos_less_divide_eq h)
```
```   838     qed
```
```   839   qed
```
```   840 qed
```
```   841
```
```   842 lemma DERIV_pos_inc_left:
```
```   843   fixes f :: "real => real"
```
```   844   shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
```
```   845   apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])
```
```   846   apply (auto simp add: DERIV_minus)
```
```   847   done
```
```   848
```
```   849 lemma DERIV_neg_dec_right:
```
```   850   fixes f :: "real => real"
```
```   851   shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
```
```   852   apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])
```
```   853   apply (auto simp add: DERIV_minus)
```
```   854   done
```
```   855
```
```   856 lemma DERIV_local_max:
```
```   857   fixes f :: "real => real"
```
```   858   assumes der: "DERIV f x :> l"
```
```   859       and d:   "0 < d"
```
```   860       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
```
```   861   shows "l = 0"
```
```   862 proof (cases rule: linorder_cases [of l 0])
```
```   863   case equal thus ?thesis .
```
```   864 next
```
```   865   case less
```
```   866   from DERIV_neg_dec_left [OF der less]
```
```   867   obtain d' where d': "0 < d'"
```
```   868              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
```
```   869   from real_lbound_gt_zero [OF d d']
```
```   870   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   871   with lt le [THEN spec [where x="x-e"]]
```
```   872   show ?thesis by (auto simp add: abs_if)
```
```   873 next
```
```   874   case greater
```
```   875   from DERIV_pos_inc_right [OF der greater]
```
```   876   obtain d' where d': "0 < d'"
```
```   877              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
```
```   878   from real_lbound_gt_zero [OF d d']
```
```   879   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   880   with lt le [THEN spec [where x="x+e"]]
```
```   881   show ?thesis by (auto simp add: abs_if)
```
```   882 qed
```
```   883
```
```   884
```
```   885 text{*Similar theorem for a local minimum*}
```
```   886 lemma DERIV_local_min:
```
```   887   fixes f :: "real => real"
```
```   888   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
```
```   889 by (drule DERIV_minus [THEN DERIV_local_max], auto)
```
```   890
```
```   891
```
```   892 text{*In particular, if a function is locally flat*}
```
```   893 lemma DERIV_local_const:
```
```   894   fixes f :: "real => real"
```
```   895   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
```
```   896 by (auto dest!: DERIV_local_max)
```
```   897
```
```   898
```
```   899 subsection {* Rolle's Theorem *}
```
```   900
```
```   901 text{*Lemma about introducing open ball in open interval*}
```
```   902 lemma lemma_interval_lt:
```
```   903      "[| a < x;  x < b |]
```
```   904       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
```
```   905
```
```   906 apply (simp add: abs_less_iff)
```
```   907 apply (insert linorder_linear [of "x-a" "b-x"], safe)
```
```   908 apply (rule_tac x = "x-a" in exI)
```
```   909 apply (rule_tac [2] x = "b-x" in exI, auto)
```
```   910 done
```
```   911
```
```   912 lemma lemma_interval: "[| a < x;  x < b |] ==>
```
```   913         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
```
```   914 apply (drule lemma_interval_lt, auto)
```
```   915 apply force
```
```   916 done
```
```   917
```
```   918 text{*Rolle's Theorem.
```
```   919    If @{term f} is defined and continuous on the closed interval
```
```   920    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
```
```   921    and @{term "f(a) = f(b)"},
```
```   922    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
```
```   923 theorem Rolle:
```
```   924   assumes lt: "a < b"
```
```   925       and eq: "f(a) = f(b)"
```
```   926       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```   927       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```   928   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
```
```   929 proof -
```
```   930   have le: "a \<le> b" using lt by simp
```
```   931   from isCont_eq_Ub [OF le con]
```
```   932   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
```
```   933              and alex: "a \<le> x" and xleb: "x \<le> b"
```
```   934     by blast
```
```   935   from isCont_eq_Lb [OF le con]
```
```   936   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
```
```   937               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
```
```   938     by blast
```
```   939   show ?thesis
```
```   940   proof cases
```
```   941     assume axb: "a < x & x < b"
```
```   942         --{*@{term f} attains its maximum within the interval*}
```
```   943     hence ax: "a<x" and xb: "x<b" by arith +
```
```   944     from lemma_interval [OF ax xb]
```
```   945     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   946       by blast
```
```   947     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
```
```   948       by blast
```
```   949     from differentiableD [OF dif [OF axb]]
```
```   950     obtain l where der: "DERIV f x :> l" ..
```
```   951     have "l=0" by (rule DERIV_local_max [OF der d bound'])
```
```   952         --{*the derivative at a local maximum is zero*}
```
```   953     thus ?thesis using ax xb der by auto
```
```   954   next
```
```   955     assume notaxb: "~ (a < x & x < b)"
```
```   956     hence xeqab: "x=a | x=b" using alex xleb by arith
```
```   957     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
```
```   958     show ?thesis
```
```   959     proof cases
```
```   960       assume ax'b: "a < x' & x' < b"
```
```   961         --{*@{term f} attains its minimum within the interval*}
```
```   962       hence ax': "a<x'" and x'b: "x'<b" by arith+
```
```   963       from lemma_interval [OF ax' x'b]
```
```   964       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   965   by blast
```
```   966       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
```
```   967   by blast
```
```   968       from differentiableD [OF dif [OF ax'b]]
```
```   969       obtain l where der: "DERIV f x' :> l" ..
```
```   970       have "l=0" by (rule DERIV_local_min [OF der d bound'])
```
```   971         --{*the derivative at a local minimum is zero*}
```
```   972       thus ?thesis using ax' x'b der by auto
```
```   973     next
```
```   974       assume notax'b: "~ (a < x' & x' < b)"
```
```   975         --{*@{term f} is constant througout the interval*}
```
```   976       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
```
```   977       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
```
```   978       from dense [OF lt]
```
```   979       obtain r where ar: "a < r" and rb: "r < b" by blast
```
```   980       from lemma_interval [OF ar rb]
```
```   981       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   982   by blast
```
```   983       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
```
```   984       proof (clarify)
```
```   985         fix z::real
```
```   986         assume az: "a \<le> z" and zb: "z \<le> b"
```
```   987         show "f z = f b"
```
```   988         proof (rule order_antisym)
```
```   989           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
```
```   990           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
```
```   991         qed
```
```   992       qed
```
```   993       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
```
```   994       proof (intro strip)
```
```   995         fix y::real
```
```   996         assume lt: "\<bar>r-y\<bar> < d"
```
```   997         hence "f y = f b" by (simp add: eq_fb bound)
```
```   998         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
```
```   999       qed
```
```  1000       from differentiableD [OF dif [OF conjI [OF ar rb]]]
```
```  1001       obtain l where der: "DERIV f r :> l" ..
```
```  1002       have "l=0" by (rule DERIV_local_const [OF der d bound'])
```
```  1003         --{*the derivative of a constant function is zero*}
```
```  1004       thus ?thesis using ar rb der by auto
```
```  1005     qed
```
```  1006   qed
```
```  1007 qed
```
```  1008
```
```  1009
```
```  1010 subsection{*Mean Value Theorem*}
```
```  1011
```
```  1012 lemma lemma_MVT:
```
```  1013      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
```
```  1014   by (cases "a = b") (simp_all add: field_simps)
```
```  1015
```
```  1016 theorem MVT:
```
```  1017   assumes lt:  "a < b"
```
```  1018       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```  1019       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```  1020   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
```
```  1021                    (f(b) - f(a) = (b-a) * l)"
```
```  1022 proof -
```
```  1023   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
```
```  1024   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
```
```  1025     using con by (fast intro: isCont_intros)
```
```  1026   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
```
```  1027   proof (clarify)
```
```  1028     fix x::real
```
```  1029     assume ax: "a < x" and xb: "x < b"
```
```  1030     from differentiableD [OF dif [OF conjI [OF ax xb]]]
```
```  1031     obtain l where der: "DERIV f x :> l" ..
```
```  1032     show "?F differentiable x"
```
```  1033       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
```
```  1034           blast intro: DERIV_diff DERIV_cmult_Id der)
```
```  1035   qed
```
```  1036   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
```
```  1037   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
```
```  1038     by blast
```
```  1039   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
```
```  1040     by (rule DERIV_cmult_Id)
```
```  1041   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
```
```  1042                    :> 0 + (f b - f a) / (b - a)"
```
```  1043     by (rule DERIV_add [OF der])
```
```  1044   show ?thesis
```
```  1045   proof (intro exI conjI)
```
```  1046     show "a < z" using az .
```
```  1047     show "z < b" using zb .
```
```  1048     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
```
```  1049     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
```
```  1050   qed
```
```  1051 qed
```
```  1052
```
```  1053 lemma MVT2:
```
```  1054      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
```
```  1055       ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
```
```  1056 apply (drule MVT)
```
```  1057 apply (blast intro: DERIV_isCont)
```
```  1058 apply (force dest: order_less_imp_le simp add: differentiable_def)
```
```  1059 apply (blast dest: DERIV_unique order_less_imp_le)
```
```  1060 done
```
```  1061
```
```  1062
```
```  1063 text{*A function is constant if its derivative is 0 over an interval.*}
```
```  1064
```
```  1065 lemma DERIV_isconst_end:
```
```  1066   fixes f :: "real => real"
```
```  1067   shows "[| a < b;
```
```  1068          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1069          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1070         ==> f b = f a"
```
```  1071 apply (drule MVT, assumption)
```
```  1072 apply (blast intro: differentiableI)
```
```  1073 apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
```
```  1074 done
```
```  1075
```
```  1076 lemma DERIV_isconst1:
```
```  1077   fixes f :: "real => real"
```
```  1078   shows "[| a < b;
```
```  1079          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1080          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1081         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
```
```  1082 apply safe
```
```  1083 apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
```
```  1084 apply (drule_tac b = x in DERIV_isconst_end, auto)
```
```  1085 done
```
```  1086
```
```  1087 lemma DERIV_isconst2:
```
```  1088   fixes f :: "real => real"
```
```  1089   shows "[| a < b;
```
```  1090          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1091          \<forall>x. a < x & x < b --> DERIV f x :> 0;
```
```  1092          a \<le> x; x \<le> b |]
```
```  1093         ==> f x = f a"
```
```  1094 apply (blast dest: DERIV_isconst1)
```
```  1095 done
```
```  1096
```
```  1097 lemma DERIV_isconst3: fixes a b x y :: real
```
```  1098   assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
```
```  1099   assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
```
```  1100   shows "f x = f y"
```
```  1101 proof (cases "x = y")
```
```  1102   case False
```
```  1103   let ?a = "min x y"
```
```  1104   let ?b = "max x y"
```
```  1105
```
```  1106   have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
```
```  1107   proof (rule allI, rule impI)
```
```  1108     fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
```
```  1109     hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
```
```  1110     hence "z \<in> {a<..<b}" by auto
```
```  1111     thus "DERIV f z :> 0" by (rule derivable)
```
```  1112   qed
```
```  1113   hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
```
```  1114     and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
```
```  1115
```
```  1116   have "?a < ?b" using `x \<noteq> y` by auto
```
```  1117   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
```
```  1118   show ?thesis by auto
```
```  1119 qed auto
```
```  1120
```
```  1121 lemma DERIV_isconst_all:
```
```  1122   fixes f :: "real => real"
```
```  1123   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
```
```  1124 apply (rule linorder_cases [of x y])
```
```  1125 apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
```
```  1126 done
```
```  1127
```
```  1128 lemma DERIV_const_ratio_const:
```
```  1129   fixes f :: "real => real"
```
```  1130   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
```
```  1131 apply (rule linorder_cases [of a b], auto)
```
```  1132 apply (drule_tac [!] f = f in MVT)
```
```  1133 apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
```
```  1134 apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
```
```  1135 done
```
```  1136
```
```  1137 lemma DERIV_const_ratio_const2:
```
```  1138   fixes f :: "real => real"
```
```  1139   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
```
```  1140 apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
```
```  1141 apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
```
```  1142 done
```
```  1143
```
```  1144 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
```
```  1145 by (simp)
```
```  1146
```
```  1147 lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
```
```  1148 by (simp)
```
```  1149
```
```  1150 text{*Gallileo's "trick": average velocity = av. of end velocities*}
```
```  1151
```
```  1152 lemma DERIV_const_average:
```
```  1153   fixes v :: "real => real"
```
```  1154   assumes neq: "a \<noteq> (b::real)"
```
```  1155       and der: "\<forall>x. DERIV v x :> k"
```
```  1156   shows "v ((a + b)/2) = (v a + v b)/2"
```
```  1157 proof (cases rule: linorder_cases [of a b])
```
```  1158   case equal with neq show ?thesis by simp
```
```  1159 next
```
```  1160   case less
```
```  1161   have "(v b - v a) / (b - a) = k"
```
```  1162     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1163   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1164   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
```
```  1165     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1166   ultimately show ?thesis using neq by force
```
```  1167 next
```
```  1168   case greater
```
```  1169   have "(v b - v a) / (b - a) = k"
```
```  1170     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1171   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1172   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
```
```  1173     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1174   ultimately show ?thesis using neq by (force simp add: add_commute)
```
```  1175 qed
```
```  1176
```
```  1177 (* A function with positive derivative is increasing.
```
```  1178    A simple proof using the MVT, by Jeremy Avigad. And variants.
```
```  1179 *)
```
```  1180 lemma DERIV_pos_imp_increasing:
```
```  1181   fixes a::real and b::real and f::"real => real"
```
```  1182   assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
```
```  1183   shows "f a < f b"
```
```  1184 proof (rule ccontr)
```
```  1185   assume f: "~ f a < f b"
```
```  1186   have "EX l z. a < z & z < b & DERIV f z :> l
```
```  1187       & f b - f a = (b - a) * l"
```
```  1188     apply (rule MVT)
```
```  1189       using assms
```
```  1190       apply auto
```
```  1191       apply (metis DERIV_isCont)
```
```  1192      apply (metis differentiableI less_le)
```
```  1193     done
```
```  1194   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
```
```  1195       and "f b - f a = (b - a) * l"
```
```  1196     by auto
```
```  1197   with assms f have "~(l > 0)"
```
```  1198     by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
```
```  1199   with assms z show False
```
```  1200     by (metis DERIV_unique less_le)
```
```  1201 qed
```
```  1202
```
```  1203 lemma DERIV_nonneg_imp_nondecreasing:
```
```  1204   fixes a::real and b::real and f::"real => real"
```
```  1205   assumes "a \<le> b" and
```
```  1206     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
```
```  1207   shows "f a \<le> f b"
```
```  1208 proof (rule ccontr, cases "a = b")
```
```  1209   assume "~ f a \<le> f b" and "a = b"
```
```  1210   then show False by auto
```
```  1211 next
```
```  1212   assume A: "~ f a \<le> f b"
```
```  1213   assume B: "a ~= b"
```
```  1214   with assms have "EX l z. a < z & z < b & DERIV f z :> l
```
```  1215       & f b - f a = (b - a) * l"
```
```  1216     apply -
```
```  1217     apply (rule MVT)
```
```  1218       apply auto
```
```  1219       apply (metis DERIV_isCont)
```
```  1220      apply (metis differentiableI less_le)
```
```  1221     done
```
```  1222   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
```
```  1223       and C: "f b - f a = (b - a) * l"
```
```  1224     by auto
```
```  1225   with A have "a < b" "f b < f a" by auto
```
```  1226   with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
```
```  1227     (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
```
```  1228   with assms z show False
```
```  1229     by (metis DERIV_unique order_less_imp_le)
```
```  1230 qed
```
```  1231
```
```  1232 lemma DERIV_neg_imp_decreasing:
```
```  1233   fixes a::real and b::real and f::"real => real"
```
```  1234   assumes "a < b" and
```
```  1235     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
```
```  1236   shows "f a > f b"
```
```  1237 proof -
```
```  1238   have "(%x. -f x) a < (%x. -f x) b"
```
```  1239     apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
```
```  1240     using assms
```
```  1241     apply auto
```
```  1242     apply (metis DERIV_minus neg_0_less_iff_less)
```
```  1243     done
```
```  1244   thus ?thesis
```
```  1245     by simp
```
```  1246 qed
```
```  1247
```
```  1248 lemma DERIV_nonpos_imp_nonincreasing:
```
```  1249   fixes a::real and b::real and f::"real => real"
```
```  1250   assumes "a \<le> b" and
```
```  1251     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
```
```  1252   shows "f a \<ge> f b"
```
```  1253 proof -
```
```  1254   have "(%x. -f x) a \<le> (%x. -f x) b"
```
```  1255     apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
```
```  1256     using assms
```
```  1257     apply auto
```
```  1258     apply (metis DERIV_minus neg_0_le_iff_le)
```
```  1259     done
```
```  1260   thus ?thesis
```
```  1261     by simp
```
```  1262 qed
```
```  1263
```
```  1264 text {* Derivative of inverse function *}
```
```  1265
```
```  1266 lemma DERIV_inverse_function:
```
```  1267   fixes f g :: "real \<Rightarrow> real"
```
```  1268   assumes der: "DERIV f (g x) :> D"
```
```  1269   assumes neq: "D \<noteq> 0"
```
```  1270   assumes a: "a < x" and b: "x < b"
```
```  1271   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
```
```  1272   assumes cont: "isCont g x"
```
```  1273   shows "DERIV g x :> inverse D"
```
```  1274 unfolding DERIV_iff2
```
```  1275 proof (rule LIM_equal2)
```
```  1276   show "0 < min (x - a) (b - x)"
```
```  1277     using a b by arith
```
```  1278 next
```
```  1279   fix y
```
```  1280   assume "norm (y - x) < min (x - a) (b - x)"
```
```  1281   hence "a < y" and "y < b"
```
```  1282     by (simp_all add: abs_less_iff)
```
```  1283   thus "(g y - g x) / (y - x) =
```
```  1284         inverse ((f (g y) - x) / (g y - g x))"
```
```  1285     by (simp add: inj)
```
```  1286 next
```
```  1287   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
```
```  1288     by (rule der [unfolded DERIV_iff2])
```
```  1289   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
```
```  1290     using inj a b by simp
```
```  1291   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
```
```  1292   proof (safe intro!: exI)
```
```  1293     show "0 < min (x - a) (b - x)"
```
```  1294       using a b by simp
```
```  1295   next
```
```  1296     fix y
```
```  1297     assume "norm (y - x) < min (x - a) (b - x)"
```
```  1298     hence y: "a < y" "y < b"
```
```  1299       by (simp_all add: abs_less_iff)
```
```  1300     assume "g y = g x"
```
```  1301     hence "f (g y) = f (g x)" by simp
```
```  1302     hence "y = x" using inj y a b by simp
```
```  1303     also assume "y \<noteq> x"
```
```  1304     finally show False by simp
```
```  1305   qed
```
```  1306   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
```
```  1307     using cont 1 2 by (rule isCont_LIM_compose2)
```
```  1308   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
```
```  1309         -- x --> inverse D"
```
```  1310     using neq by (rule tendsto_inverse)
```
```  1311 qed
```
```  1312
```
```  1313 subsection {* Generalized Mean Value Theorem *}
```
```  1314
```
```  1315 theorem GMVT:
```
```  1316   fixes a b :: real
```
```  1317   assumes alb: "a < b"
```
```  1318     and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  1319     and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
```
```  1320     and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
```
```  1321     and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
```
```  1322   shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
```
```  1323 proof -
```
```  1324   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
```
```  1325   from assms have "a < b" by simp
```
```  1326   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
```
```  1327     using fc gc by simp
```
```  1328   moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
```
```  1329     using fd gd by simp
```
```  1330   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)
```
```  1331   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" ..
```
```  1332   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
```
```  1333
```
```  1334   from cdef have cint: "a < c \<and> c < b" by auto
```
```  1335   with gd have "g differentiable c" by simp
```
```  1336   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
```
```  1337   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
```
```  1338
```
```  1339   from cdef have "a < c \<and> c < b" by auto
```
```  1340   with fd have "f differentiable c" by simp
```
```  1341   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
```
```  1342   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
```
```  1343
```
```  1344   from cdef have "DERIV ?h c :> l" by auto
```
```  1345   moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
```
```  1346     using g'cdef f'cdef by (auto intro!: DERIV_intros)
```
```  1347   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
```
```  1348
```
```  1349   {
```
```  1350     from cdef have "?h b - ?h a = (b - a) * l" by auto
```
```  1351     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1352     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1353   }
```
```  1354   moreover
```
```  1355   {
```
```  1356     have "?h b - ?h a =
```
```  1357          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
```
```  1358           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
```
```  1359       by (simp add: algebra_simps)
```
```  1360     hence "?h b - ?h a = 0" by auto
```
```  1361   }
```
```  1362   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
```
```  1363   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
```
```  1364   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
```
```  1365   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
```
```  1366
```
```  1367   with g'cdef f'cdef cint show ?thesis by auto
```
```  1368 qed
```
```  1369
```
```  1370 lemma GMVT':
```
```  1371   fixes f g :: "real \<Rightarrow> real"
```
```  1372   assumes "a < b"
```
```  1373   assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
```
```  1374   assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
```
```  1375   assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
```
```  1376   assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
```
```  1377   shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
```
```  1378 proof -
```
```  1379   have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
```
```  1380     a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
```
```  1381     using assms by (intro GMVT) (force simp: differentiable_def)+
```
```  1382   then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
```
```  1383     using DERIV_f DERIV_g by (force dest: DERIV_unique)
```
```  1384   then show ?thesis
```
```  1385     by auto
```
```  1386 qed
```
```  1387
```
```  1388
```
```  1389 subsection {* L'Hopitals rule *}
```
```  1390
```
```  1391 lemma isCont_If_ge:
```
```  1392   fixes a :: "'a :: linorder_topology"
```
```  1393   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"
```
```  1394   unfolding isCont_def continuous_within
```
```  1395   apply (intro filterlim_split_at)
```
```  1396   apply (subst filterlim_cong[OF refl refl, where g=g])
```
```  1397   apply (simp_all add: eventually_at_filter less_le)
```
```  1398   apply (subst filterlim_cong[OF refl refl, where g=f])
```
```  1399   apply (simp_all add: eventually_at_filter less_le)
```
```  1400   done
```
```  1401
```
```  1402 lemma lhopital_right_0:
```
```  1403   fixes f0 g0 :: "real \<Rightarrow> real"
```
```  1404   assumes f_0: "(f0 ---> 0) (at_right 0)"
```
```  1405   assumes g_0: "(g0 ---> 0) (at_right 0)"
```
```  1406   assumes ev:
```
```  1407     "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
```
```  1408     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
```
```  1409     "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
```
```  1410     "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
```
```  1411   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
```
```  1412   shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
```
```  1413 proof -
```
```  1414   def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
```
```  1415   then have "f 0 = 0" by simp
```
```  1416
```
```  1417   def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
```
```  1418   then have "g 0 = 0" by simp
```
```  1419
```
```  1420   have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
```
```  1421       DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
```
```  1422     using ev by eventually_elim auto
```
```  1423   then obtain a where [arith]: "0 < a"
```
```  1424     and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
```
```  1425     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
```
```  1426     and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
```
```  1427     and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
```
```  1428     unfolding eventually_at eventually_at by (auto simp: dist_real_def)
```
```  1429
```
```  1430   have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
```
```  1431     using g0_neq_0 by (simp add: g_def)
```
```  1432
```
```  1433   { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
```
```  1434       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
```
```  1435          (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
```
```  1436   note f = this
```
```  1437
```
```  1438   { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
```
```  1439       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
```
```  1440          (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
```
```  1441   note g = this
```
```  1442
```
```  1443   have "isCont f 0"
```
```  1444     unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
```
```  1445
```
```  1446   have "isCont g 0"
```
```  1447     unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
```
```  1448
```
```  1449   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)"
```
```  1450   proof (rule bchoice, rule)
```
```  1451     fix x assume "x \<in> {0 <..< a}"
```
```  1452     then have x[arith]: "0 < x" "x < a" by auto
```
```  1453     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"
```
```  1454       by auto
```
```  1455     have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
```
```  1456       using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
```
```  1457     moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
```
```  1458       using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
```
```  1459     ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
```
```  1460       using f g `x < a` by (intro GMVT') auto
```
```  1461     then guess c ..
```
```  1462     moreover
```
```  1463     with g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
```
```  1464       by (simp add: field_simps)
```
```  1465     ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
```
```  1466       using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
```
```  1467   qed
```
```  1468   then guess \<zeta> ..
```
```  1469   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)"
```
```  1470     unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
```
```  1471   moreover
```
```  1472   from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
```
```  1473     by eventually_elim auto
```
```  1474   then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
```
```  1475     by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
```
```  1476        (auto intro: tendsto_const tendsto_ident_at)
```
```  1477   then have "(\<zeta> ---> 0) (at_right 0)"
```
```  1478     by (rule tendsto_norm_zero_cancel)
```
```  1479   with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
```
```  1480     by (auto elim!: eventually_elim1 simp: filterlim_at)
```
```  1481   from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
```
```  1482     by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
```
```  1483   ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
```
```  1484     by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
```
```  1485        (auto elim: eventually_elim1)
```
```  1486   also have "?P \<longleftrightarrow> ?thesis"
```
```  1487     by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
```
```  1488   finally show ?thesis .
```
```  1489 qed
```
```  1490
```
```  1491 lemma lhopital_right:
```
```  1492   "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
```
```  1493     eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1494     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1495     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
```
```  1496     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
```
```  1497     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
```
```  1498   ((\<lambda> x. f x / g x) ---> y) (at_right x)"
```
```  1499   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
```
```  1500   by (rule lhopital_right_0)
```
```  1501
```
```  1502 lemma lhopital_left:
```
```  1503   "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
```
```  1504     eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1505     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1506     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
```
```  1507     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
```
```  1508     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
```
```  1509   ((\<lambda> x. f x / g x) ---> y) (at_left x)"
```
```  1510   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
```
```  1511   by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
```
```  1512
```
```  1513 lemma lhopital:
```
```  1514   "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
```
```  1515     eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1516     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1517     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
```
```  1518     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
```
```  1519     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
```
```  1520   ((\<lambda> x. f x / g x) ---> y) (at x)"
```
```  1521   unfolding eventually_at_split filterlim_at_split
```
```  1522   by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
```
```  1523
```
```  1524 lemma lhopital_right_0_at_top:
```
```  1525   fixes f g :: "real \<Rightarrow> real"
```
```  1526   assumes g_0: "LIM x at_right 0. g x :> at_top"
```
```  1527   assumes ev:
```
```  1528     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
```
```  1529     "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
```
```  1530     "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
```
```  1531   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
```
```  1532   shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
```
```  1533   unfolding tendsto_iff
```
```  1534 proof safe
```
```  1535   fix e :: real assume "0 < e"
```
```  1536
```
```  1537   with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
```
```  1538   have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
```
```  1539   from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
```
```  1540   obtain a where [arith]: "0 < a"
```
```  1541     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
```
```  1542     and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
```
```  1543     and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
```
```  1544     and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
```
```  1545     unfolding eventually_at_le by (auto simp: dist_real_def)
```
```  1546
```
```  1547
```
```  1548   from Df have
```
```  1549     "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
```
```  1550     unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
```
```  1551
```
```  1552   moreover
```
```  1553   have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
```
```  1554     using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
```
```  1555
```
```  1556   moreover
```
```  1557   have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
```
```  1558     using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
```
```  1559     by (rule filterlim_compose)
```
```  1560   then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
```
```  1561     by (intro tendsto_intros)
```
```  1562   then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
```
```  1563     by (simp add: inverse_eq_divide)
```
```  1564   from this[unfolded tendsto_iff, rule_format, of 1]
```
```  1565   have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
```
```  1566     by (auto elim!: eventually_elim1 simp: dist_real_def)
```
```  1567
```
```  1568   moreover
```
```  1569   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)"
```
```  1570     by (intro tendsto_intros)
```
```  1571   then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
```
```  1572     by (simp add: inverse_eq_divide)
```
```  1573   from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
```
```  1574   have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
```
```  1575     by (auto simp: dist_real_def)
```
```  1576
```
```  1577   ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
```
```  1578   proof eventually_elim
```
```  1579     fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
```
```  1580     assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
```
```  1581
```
```  1582     have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
```
```  1583       using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
```
```  1584     then guess y ..
```
```  1585     from this
```
```  1586     have [arith]: "t < y" "y < a" and D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
```
```  1587       using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
```
```  1588
```
```  1589     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"
```
```  1590       by (simp add: field_simps)
```
```  1591     have "norm (f t / g t - x) \<le>
```
```  1592         norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
```
```  1593       unfolding * by (rule norm_triangle_ineq)
```
```  1594     also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
```
```  1595       by (simp add: abs_mult D_eq dist_real_def)
```
```  1596     also have "\<dots> < (e / 4) * 2 + e / 2"
```
```  1597       using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
```
```  1598     finally show "dist (f t / g t) x < e"
```
```  1599       by (simp add: dist_real_def)
```
```  1600   qed
```
```  1601 qed
```
```  1602
```
```  1603 lemma lhopital_right_at_top:
```
```  1604   "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1605     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1606     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
```
```  1607     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
```
```  1608     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
```
```  1609     ((\<lambda> x. f x / g x) ---> y) (at_right x)"
```
```  1610   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
```
```  1611   by (rule lhopital_right_0_at_top)
```
```  1612
```
```  1613 lemma lhopital_left_at_top:
```
```  1614   "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1615     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1616     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
```
```  1617     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
```
```  1618     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
```
```  1619     ((\<lambda> x. f x / g x) ---> y) (at_left x)"
```
```  1620   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
```
```  1621   by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
```
```  1622
```
```  1623 lemma lhopital_at_top:
```
```  1624   "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1625     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1626     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
```
```  1627     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
```
```  1628     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
```
```  1629     ((\<lambda> x. f x / g x) ---> y) (at x)"
```
```  1630   unfolding eventually_at_split filterlim_at_split
```
```  1631   by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
```
```  1632
```
```  1633 lemma lhospital_at_top_at_top:
```
```  1634   fixes f g :: "real \<Rightarrow> real"
```
```  1635   assumes g_0: "LIM x at_top. g x :> at_top"
```
```  1636   assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
```
```  1637   assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
```
```  1638   assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
```
```  1639   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
```
```  1640   shows "((\<lambda> x. f x / g x) ---> x) at_top"
```
```  1641   unfolding filterlim_at_top_to_right
```
```  1642 proof (rule lhopital_right_0_at_top)
```
```  1643   let ?F = "\<lambda>x. f (inverse x)"
```
```  1644   let ?G = "\<lambda>x. g (inverse x)"
```
```  1645   let ?R = "at_right (0::real)"
```
```  1646   let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
```
```  1647
```
```  1648   show "LIM x ?R. ?G x :> at_top"
```
```  1649     using g_0 unfolding filterlim_at_top_to_right .
```
```  1650
```
```  1651   show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
```
```  1652     unfolding eventually_at_right_to_top
```
```  1653     using Dg eventually_ge_at_top[where c="1::real"]
```
```  1654     apply eventually_elim
```
```  1655     apply (rule DERIV_cong)
```
```  1656     apply (rule DERIV_chain'[where f=inverse])
```
```  1657     apply (auto intro!:  DERIV_inverse)
```
```  1658     done
```
```  1659
```
```  1660   show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
```
```  1661     unfolding eventually_at_right_to_top
```
```  1662     using Df eventually_ge_at_top[where c="1::real"]
```
```  1663     apply eventually_elim
```
```  1664     apply (rule DERIV_cong)
```
```  1665     apply (rule DERIV_chain'[where f=inverse])
```
```  1666     apply (auto intro!:  DERIV_inverse)
```
```  1667     done
```
```  1668
```
```  1669   show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
```
```  1670     unfolding eventually_at_right_to_top
```
```  1671     using g' eventually_ge_at_top[where c="1::real"]
```
```  1672     by eventually_elim auto
```
```  1673
```
```  1674   show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
```
```  1675     unfolding filterlim_at_right_to_top
```
```  1676     apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
```
```  1677     using eventually_ge_at_top[where c="1::real"]
```
```  1678     by eventually_elim simp
```
```  1679 qed
```
```  1680
```
```  1681 end
```