src/HOL/Library/FrechetDeriv.thy
 author wenzelm Thu Feb 11 23:00:22 2010 +0100 (2010-02-11) changeset 35115 446c5063e4fd parent 34146 14595e0c27e8 child 36361 1debc8e29f6a permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;
```     1 (*  Title       : FrechetDeriv.thy
```
```     2     Author      : Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Frechet Derivative *}
```
```     6
```
```     7 theory FrechetDeriv
```
```     8 imports Lim Complex_Main
```
```     9 begin
```
```    10
```
```    11 definition
```
```    12   fderiv ::
```
```    13   "['a::real_normed_vector \<Rightarrow> 'b::real_normed_vector, 'a, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
```
```    14     -- {* Frechet derivative: D is derivative of function f at x *}
```
```    15           ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
```
```    16   "FDERIV f x :> D = (bounded_linear D \<and>
```
```    17     (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0)"
```
```    18
```
```    19 lemma FDERIV_I:
```
```    20   "\<lbrakk>bounded_linear D; (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0\<rbrakk>
```
```    21    \<Longrightarrow> FDERIV f x :> D"
```
```    22 by (simp add: fderiv_def)
```
```    23
```
```    24 lemma FDERIV_D:
```
```    25   "FDERIV f x :> D \<Longrightarrow> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0"
```
```    26 by (simp add: fderiv_def)
```
```    27
```
```    28 lemma FDERIV_bounded_linear: "FDERIV f x :> D \<Longrightarrow> bounded_linear D"
```
```    29 by (simp add: fderiv_def)
```
```    30
```
```    31 lemma bounded_linear_zero:
```
```    32   "bounded_linear (\<lambda>x::'a::real_normed_vector. 0::'b::real_normed_vector)"
```
```    33 proof
```
```    34   show "(0::'b) = 0 + 0" by simp
```
```    35   fix r show "(0::'b) = scaleR r 0" by simp
```
```    36   have "\<forall>x::'a. norm (0::'b) \<le> norm x * 0" by simp
```
```    37   thus "\<exists>K. \<forall>x::'a. norm (0::'b) \<le> norm x * K" ..
```
```    38 qed
```
```    39
```
```    40 lemma FDERIV_const: "FDERIV (\<lambda>x. k) x :> (\<lambda>h. 0)"
```
```    41 by (simp add: fderiv_def bounded_linear_zero)
```
```    42
```
```    43 lemma bounded_linear_ident:
```
```    44   "bounded_linear (\<lambda>x::'a::real_normed_vector. x)"
```
```    45 proof
```
```    46   fix x y :: 'a show "x + y = x + y" by simp
```
```    47   fix r and x :: 'a show "scaleR r x = scaleR r x" by simp
```
```    48   have "\<forall>x::'a. norm x \<le> norm x * 1" by simp
```
```    49   thus "\<exists>K. \<forall>x::'a. norm x \<le> norm x * K" ..
```
```    50 qed
```
```    51
```
```    52 lemma FDERIV_ident: "FDERIV (\<lambda>x. x) x :> (\<lambda>h. h)"
```
```    53 by (simp add: fderiv_def bounded_linear_ident)
```
```    54
```
```    55 subsection {* Addition *}
```
```    56
```
```    57 lemma add_diff_add:
```
```    58   fixes a b c d :: "'a::ab_group_add"
```
```    59   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```    60 by simp
```
```    61
```
```    62 lemma bounded_linear_add:
```
```    63   assumes "bounded_linear f"
```
```    64   assumes "bounded_linear g"
```
```    65   shows "bounded_linear (\<lambda>x. f x + g x)"
```
```    66 proof -
```
```    67   interpret f: bounded_linear f by fact
```
```    68   interpret g: bounded_linear g by fact
```
```    69   show ?thesis apply (unfold_locales)
```
```    70     apply (simp only: f.add g.add add_ac)
```
```    71     apply (simp only: f.scaleR g.scaleR scaleR_right_distrib)
```
```    72     apply (rule f.pos_bounded [THEN exE], rename_tac Kf)
```
```    73     apply (rule g.pos_bounded [THEN exE], rename_tac Kg)
```
```    74     apply (rule_tac x="Kf + Kg" in exI, safe)
```
```    75     apply (subst right_distrib)
```
```    76     apply (rule order_trans [OF norm_triangle_ineq])
```
```    77     apply (rule add_mono, erule spec, erule spec)
```
```    78     done
```
```    79 qed
```
```    80
```
```    81 lemma norm_ratio_ineq:
```
```    82   fixes x y :: "'a::real_normed_vector"
```
```    83   fixes h :: "'b::real_normed_vector"
```
```    84   shows "norm (x + y) / norm h \<le> norm x / norm h + norm y / norm h"
```
```    85 apply (rule ord_le_eq_trans)
```
```    86 apply (rule divide_right_mono)
```
```    87 apply (rule norm_triangle_ineq)
```
```    88 apply (rule norm_ge_zero)
```
```    89 apply (rule add_divide_distrib)
```
```    90 done
```
```    91
```
```    92 lemma FDERIV_add:
```
```    93   assumes f: "FDERIV f x :> F"
```
```    94   assumes g: "FDERIV g x :> G"
```
```    95   shows "FDERIV (\<lambda>x. f x + g x) x :> (\<lambda>h. F h + G h)"
```
```    96 proof (rule FDERIV_I)
```
```    97   show "bounded_linear (\<lambda>h. F h + G h)"
```
```    98     apply (rule bounded_linear_add)
```
```    99     apply (rule FDERIV_bounded_linear [OF f])
```
```   100     apply (rule FDERIV_bounded_linear [OF g])
```
```   101     done
```
```   102 next
```
```   103   have f': "(\<lambda>h. norm (f (x + h) - f x - F h) / norm h) -- 0 --> 0"
```
```   104     using f by (rule FDERIV_D)
```
```   105   have g': "(\<lambda>h. norm (g (x + h) - g x - G h) / norm h) -- 0 --> 0"
```
```   106     using g by (rule FDERIV_D)
```
```   107   from f' g'
```
```   108   have "(\<lambda>h. norm (f (x + h) - f x - F h) / norm h
```
```   109            + norm (g (x + h) - g x - G h) / norm h) -- 0 --> 0"
```
```   110     by (rule LIM_add_zero)
```
```   111   thus "(\<lambda>h. norm (f (x + h) + g (x + h) - (f x + g x) - (F h + G h))
```
```   112            / norm h) -- 0 --> 0"
```
```   113     apply (rule real_LIM_sandwich_zero)
```
```   114      apply (simp add: divide_nonneg_pos)
```
```   115     apply (simp only: add_diff_add)
```
```   116     apply (rule norm_ratio_ineq)
```
```   117     done
```
```   118 qed
```
```   119
```
```   120 subsection {* Subtraction *}
```
```   121
```
```   122 lemma bounded_linear_minus:
```
```   123   assumes "bounded_linear f"
```
```   124   shows "bounded_linear (\<lambda>x. - f x)"
```
```   125 proof -
```
```   126   interpret f: bounded_linear f by fact
```
```   127   show ?thesis apply (unfold_locales)
```
```   128     apply (simp add: f.add)
```
```   129     apply (simp add: f.scaleR)
```
```   130     apply (simp add: f.bounded)
```
```   131     done
```
```   132 qed
```
```   133
```
```   134 lemma FDERIV_minus:
```
```   135   "FDERIV f x :> F \<Longrightarrow> FDERIV (\<lambda>x. - f x) x :> (\<lambda>h. - F h)"
```
```   136 apply (rule FDERIV_I)
```
```   137 apply (rule bounded_linear_minus)
```
```   138 apply (erule FDERIV_bounded_linear)
```
```   139 apply (simp only: fderiv_def minus_diff_minus norm_minus_cancel)
```
```   140 done
```
```   141
```
```   142 lemma FDERIV_diff:
```
```   143   "\<lbrakk>FDERIV f x :> F; FDERIV g x :> G\<rbrakk>
```
```   144    \<Longrightarrow> FDERIV (\<lambda>x. f x - g x) x :> (\<lambda>h. F h - G h)"
```
```   145 by (simp only: diff_minus FDERIV_add FDERIV_minus)
```
```   146
```
```   147 subsection {* Continuity *}
```
```   148
```
```   149 lemma FDERIV_isCont:
```
```   150   assumes f: "FDERIV f x :> F"
```
```   151   shows "isCont f x"
```
```   152 proof -
```
```   153   from f interpret F: bounded_linear "F" by (rule FDERIV_bounded_linear)
```
```   154   have "(\<lambda>h. norm (f (x + h) - f x - F h) / norm h) -- 0 --> 0"
```
```   155     by (rule FDERIV_D [OF f])
```
```   156   hence "(\<lambda>h. norm (f (x + h) - f x - F h) / norm h * norm h) -- 0 --> 0"
```
```   157     by (intro LIM_mult_zero LIM_norm_zero LIM_ident)
```
```   158   hence "(\<lambda>h. norm (f (x + h) - f x - F h)) -- 0 --> 0"
```
```   159     by (simp cong: LIM_cong)
```
```   160   hence "(\<lambda>h. f (x + h) - f x - F h) -- 0 --> 0"
```
```   161     by (rule LIM_norm_zero_cancel)
```
```   162   hence "(\<lambda>h. f (x + h) - f x - F h + F h) -- 0 --> 0"
```
```   163     by (intro LIM_add_zero F.LIM_zero LIM_ident)
```
```   164   hence "(\<lambda>h. f (x + h) - f x) -- 0 --> 0"
```
```   165     by simp
```
```   166   thus "isCont f x"
```
```   167     unfolding isCont_iff by (rule LIM_zero_cancel)
```
```   168 qed
```
```   169
```
```   170 subsection {* Composition *}
```
```   171
```
```   172 lemma real_divide_cancel_lemma:
```
```   173   fixes a b c :: real
```
```   174   shows "(b = 0 \<Longrightarrow> a = 0) \<Longrightarrow> (a / b) * (b / c) = a / c"
```
```   175 by simp
```
```   176
```
```   177 lemma bounded_linear_compose:
```
```   178   assumes "bounded_linear f"
```
```   179   assumes "bounded_linear g"
```
```   180   shows "bounded_linear (\<lambda>x. f (g x))"
```
```   181 proof -
```
```   182   interpret f: bounded_linear f by fact
```
```   183   interpret g: bounded_linear g by fact
```
```   184   show ?thesis proof (unfold_locales)
```
```   185     fix x y show "f (g (x + y)) = f (g x) + f (g y)"
```
```   186       by (simp only: f.add g.add)
```
```   187   next
```
```   188     fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
```
```   189       by (simp only: f.scaleR g.scaleR)
```
```   190   next
```
```   191     from f.pos_bounded
```
```   192     obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by fast
```
```   193     from g.pos_bounded
```
```   194     obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by fast
```
```   195     show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
```
```   196     proof (intro exI allI)
```
```   197       fix x
```
```   198       have "norm (f (g x)) \<le> norm (g x) * Kf"
```
```   199         using f .
```
```   200       also have "\<dots> \<le> (norm x * Kg) * Kf"
```
```   201         using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
```
```   202       also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
```
```   203         by (rule mult_assoc)
```
```   204       finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
```
```   205     qed
```
```   206   qed
```
```   207 qed
```
```   208
```
```   209 lemma FDERIV_compose:
```
```   210   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   211   fixes g :: "'b::real_normed_vector \<Rightarrow> 'c::real_normed_vector"
```
```   212   assumes f: "FDERIV f x :> F"
```
```   213   assumes g: "FDERIV g (f x) :> G"
```
```   214   shows "FDERIV (\<lambda>x. g (f x)) x :> (\<lambda>h. G (F h))"
```
```   215 proof (rule FDERIV_I)
```
```   216   from FDERIV_bounded_linear [OF g] FDERIV_bounded_linear [OF f]
```
```   217   show "bounded_linear (\<lambda>h. G (F h))"
```
```   218     by (rule bounded_linear_compose)
```
```   219 next
```
```   220   let ?Rf = "\<lambda>h. f (x + h) - f x - F h"
```
```   221   let ?Rg = "\<lambda>k. g (f x + k) - g (f x) - G k"
```
```   222   let ?k = "\<lambda>h. f (x + h) - f x"
```
```   223   let ?Nf = "\<lambda>h. norm (?Rf h) / norm h"
```
```   224   let ?Ng = "\<lambda>h. norm (?Rg (?k h)) / norm (?k h)"
```
```   225   from f interpret F: bounded_linear "F" by (rule FDERIV_bounded_linear)
```
```   226   from g interpret G: bounded_linear "G" by (rule FDERIV_bounded_linear)
```
```   227   from F.bounded obtain kF where kF: "\<And>x. norm (F x) \<le> norm x * kF" by fast
```
```   228   from G.bounded obtain kG where kG: "\<And>x. norm (G x) \<le> norm x * kG" by fast
```
```   229
```
```   230   let ?fun2 = "\<lambda>h. ?Nf h * kG + ?Ng h * (?Nf h + kF)"
```
```   231
```
```   232   show "(\<lambda>h. norm (g (f (x + h)) - g (f x) - G (F h)) / norm h) -- 0 --> 0"
```
```   233   proof (rule real_LIM_sandwich_zero)
```
```   234     have Nf: "?Nf -- 0 --> 0"
```
```   235       using FDERIV_D [OF f] .
```
```   236
```
```   237     have Ng1: "isCont (\<lambda>k. norm (?Rg k) / norm k) 0"
```
```   238       by (simp add: isCont_def FDERIV_D [OF g])
```
```   239     have Ng2: "?k -- 0 --> 0"
```
```   240       apply (rule LIM_zero)
```
```   241       apply (fold isCont_iff)
```
```   242       apply (rule FDERIV_isCont [OF f])
```
```   243       done
```
```   244     have Ng: "?Ng -- 0 --> 0"
```
```   245       using isCont_LIM_compose [OF Ng1 Ng2] by simp
```
```   246
```
```   247     have "(\<lambda>h. ?Nf h * kG + ?Ng h * (?Nf h + kF))
```
```   248            -- 0 --> 0 * kG + 0 * (0 + kF)"
```
```   249       by (intro LIM_add LIM_mult LIM_const Nf Ng)
```
```   250     thus "(\<lambda>h. ?Nf h * kG + ?Ng h * (?Nf h + kF)) -- 0 --> 0"
```
```   251       by simp
```
```   252   next
```
```   253     fix h::'a assume h: "h \<noteq> 0"
```
```   254     thus "0 \<le> norm (g (f (x + h)) - g (f x) - G (F h)) / norm h"
```
```   255       by (simp add: divide_nonneg_pos)
```
```   256   next
```
```   257     fix h::'a assume h: "h \<noteq> 0"
```
```   258     have "g (f (x + h)) - g (f x) - G (F h) = G (?Rf h) + ?Rg (?k h)"
```
```   259       by (simp add: G.diff)
```
```   260     hence "norm (g (f (x + h)) - g (f x) - G (F h)) / norm h
```
```   261            = norm (G (?Rf h) + ?Rg (?k h)) / norm h"
```
```   262       by (rule arg_cong)
```
```   263     also have "\<dots> \<le> norm (G (?Rf h)) / norm h + norm (?Rg (?k h)) / norm h"
```
```   264       by (rule norm_ratio_ineq)
```
```   265     also have "\<dots> \<le> ?Nf h * kG + ?Ng h * (?Nf h + kF)"
```
```   266     proof (rule add_mono)
```
```   267       show "norm (G (?Rf h)) / norm h \<le> ?Nf h * kG"
```
```   268         apply (rule ord_le_eq_trans)
```
```   269         apply (rule divide_right_mono [OF kG norm_ge_zero])
```
```   270         apply simp
```
```   271         done
```
```   272     next
```
```   273       have "norm (?Rg (?k h)) / norm h = ?Ng h * (norm (?k h) / norm h)"
```
```   274         apply (rule real_divide_cancel_lemma [symmetric])
```
```   275         apply (simp add: G.zero)
```
```   276         done
```
```   277       also have "\<dots> \<le> ?Ng h * (?Nf h + kF)"
```
```   278       proof (rule mult_left_mono)
```
```   279         have "norm (?k h) / norm h = norm (?Rf h + F h) / norm h"
```
```   280           by simp
```
```   281         also have "\<dots> \<le> ?Nf h + norm (F h) / norm h"
```
```   282           by (rule norm_ratio_ineq)
```
```   283         also have "\<dots> \<le> ?Nf h + kF"
```
```   284           apply (rule add_left_mono)
```
```   285           apply (subst pos_divide_le_eq, simp add: h)
```
```   286           apply (subst mult_commute)
```
```   287           apply (rule kF)
```
```   288           done
```
```   289         finally show "norm (?k h) / norm h \<le> ?Nf h + kF" .
```
```   290       next
```
```   291         show "0 \<le> ?Ng h"
```
```   292         apply (case_tac "f (x + h) - f x = 0", simp)
```
```   293         apply (rule divide_nonneg_pos [OF norm_ge_zero])
```
```   294         apply simp
```
```   295         done
```
```   296       qed
```
```   297       finally show "norm (?Rg (?k h)) / norm h \<le> ?Ng h * (?Nf h + kF)" .
```
```   298     qed
```
```   299     finally show "norm (g (f (x + h)) - g (f x) - G (F h)) / norm h
```
```   300         \<le> ?Nf h * kG + ?Ng h * (?Nf h + kF)" .
```
```   301   qed
```
```   302 qed
```
```   303
```
```   304 subsection {* Product Rule *}
```
```   305
```
```   306 lemma (in bounded_bilinear) FDERIV_lemma:
```
```   307   "a' ** b' - a ** b - (a ** B + A ** b)
```
```   308    = a ** (b' - b - B) + (a' - a - A) ** b' + A ** (b' - b)"
```
```   309 by (simp add: diff_left diff_right)
```
```   310
```
```   311 lemma (in bounded_bilinear) FDERIV:
```
```   312   fixes x :: "'d::real_normed_vector"
```
```   313   assumes f: "FDERIV f x :> F"
```
```   314   assumes g: "FDERIV g x :> G"
```
```   315   shows "FDERIV (\<lambda>x. f x ** g x) x :> (\<lambda>h. f x ** G h + F h ** g x)"
```
```   316 proof (rule FDERIV_I)
```
```   317   show "bounded_linear (\<lambda>h. f x ** G h + F h ** g x)"
```
```   318     apply (rule bounded_linear_add)
```
```   319     apply (rule bounded_linear_compose [OF bounded_linear_right])
```
```   320     apply (rule FDERIV_bounded_linear [OF g])
```
```   321     apply (rule bounded_linear_compose [OF bounded_linear_left])
```
```   322     apply (rule FDERIV_bounded_linear [OF f])
```
```   323     done
```
```   324 next
```
```   325   from bounded_linear.bounded [OF FDERIV_bounded_linear [OF f]]
```
```   326   obtain KF where norm_F: "\<And>x. norm (F x) \<le> norm x * KF" by fast
```
```   327
```
```   328   from pos_bounded obtain K where K: "0 < K" and norm_prod:
```
```   329     "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
```
```   330
```
```   331   let ?Rf = "\<lambda>h. f (x + h) - f x - F h"
```
```   332   let ?Rg = "\<lambda>h. g (x + h) - g x - G h"
```
```   333
```
```   334   let ?fun1 = "\<lambda>h.
```
```   335         norm (f x ** ?Rg h + ?Rf h ** g (x + h) + F h ** (g (x + h) - g x)) /
```
```   336         norm h"
```
```   337
```
```   338   let ?fun2 = "\<lambda>h.
```
```   339         norm (f x) * (norm (?Rg h) / norm h) * K +
```
```   340         norm (?Rf h) / norm h * norm (g (x + h)) * K +
```
```   341         KF * norm (g (x + h) - g x) * K"
```
```   342
```
```   343   have "?fun1 -- 0 --> 0"
```
```   344   proof (rule real_LIM_sandwich_zero)
```
```   345     from f g isCont_iff [THEN iffD1, OF FDERIV_isCont [OF g]]
```
```   346     have "?fun2 -- 0 -->
```
```   347           norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K"
```
```   348       by (intro LIM_add LIM_mult LIM_const LIM_norm LIM_zero FDERIV_D)
```
```   349     thus "?fun2 -- 0 --> 0"
```
```   350       by simp
```
```   351   next
```
```   352     fix h::'d assume "h \<noteq> 0"
```
```   353     thus "0 \<le> ?fun1 h"
```
```   354       by (simp add: divide_nonneg_pos)
```
```   355   next
```
```   356     fix h::'d assume "h \<noteq> 0"
```
```   357     have "?fun1 h \<le> (norm (f x) * norm (?Rg h) * K +
```
```   358          norm (?Rf h) * norm (g (x + h)) * K +
```
```   359          norm h * KF * norm (g (x + h) - g x) * K) / norm h"
```
```   360       by (intro
```
```   361         divide_right_mono mult_mono'
```
```   362         order_trans [OF norm_triangle_ineq add_mono]
```
```   363         order_trans [OF norm_prod mult_right_mono]
```
```   364         mult_nonneg_nonneg order_refl norm_ge_zero norm_F
```
```   365         K [THEN order_less_imp_le]
```
```   366       )
```
```   367     also have "\<dots> = ?fun2 h"
```
```   368       by (simp add: add_divide_distrib)
```
```   369     finally show "?fun1 h \<le> ?fun2 h" .
```
```   370   qed
```
```   371   thus "(\<lambda>h.
```
```   372     norm (f (x + h) ** g (x + h) - f x ** g x - (f x ** G h + F h ** g x))
```
```   373     / norm h) -- 0 --> 0"
```
```   374     by (simp only: FDERIV_lemma)
```
```   375 qed
```
```   376
```
```   377 lemmas FDERIV_mult = mult.FDERIV
```
```   378
```
```   379 lemmas FDERIV_scaleR = scaleR.FDERIV
```
```   380
```
```   381
```
```   382 subsection {* Powers *}
```
```   383
```
```   384 lemma FDERIV_power_Suc:
```
```   385   fixes x :: "'a::{real_normed_algebra,comm_ring_1}"
```
```   386   shows "FDERIV (\<lambda>x. x ^ Suc n) x :> (\<lambda>h. (1 + of_nat n) * x ^ n * h)"
```
```   387  apply (induct n)
```
```   388   apply (simp add: power_Suc FDERIV_ident)
```
```   389  apply (drule FDERIV_mult [OF FDERIV_ident])
```
```   390  apply (simp only: of_nat_Suc left_distrib mult_1_left)
```
```   391  apply (simp only: power_Suc right_distrib add_ac mult_ac)
```
```   392 done
```
```   393
```
```   394 lemma FDERIV_power:
```
```   395   fixes x :: "'a::{real_normed_algebra,comm_ring_1}"
```
```   396   shows "FDERIV (\<lambda>x. x ^ n) x :> (\<lambda>h. of_nat n * x ^ (n - 1) * h)"
```
```   397   apply (cases n)
```
```   398    apply (simp add: FDERIV_const)
```
```   399   apply (simp add: FDERIV_power_Suc del: power_Suc)
```
```   400   done
```
```   401
```
```   402
```
```   403 subsection {* Inverse *}
```
```   404
```
```   405 lemma inverse_diff_inverse:
```
```   406   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   407    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   408 by (simp add: right_diff_distrib left_diff_distrib mult_assoc)
```
```   409
```
```   410 lemmas bounded_linear_mult_const =
```
```   411   mult.bounded_linear_left [THEN bounded_linear_compose]
```
```   412
```
```   413 lemmas bounded_linear_const_mult =
```
```   414   mult.bounded_linear_right [THEN bounded_linear_compose]
```
```   415
```
```   416 lemma FDERIV_inverse:
```
```   417   fixes x :: "'a::real_normed_div_algebra"
```
```   418   assumes x: "x \<noteq> 0"
```
```   419   shows "FDERIV inverse x :> (\<lambda>h. - (inverse x * h * inverse x))"
```
```   420         (is "FDERIV ?inv _ :> _")
```
```   421 proof (rule FDERIV_I)
```
```   422   show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
```
```   423     apply (rule bounded_linear_minus)
```
```   424     apply (rule bounded_linear_mult_const)
```
```   425     apply (rule bounded_linear_const_mult)
```
```   426     apply (rule bounded_linear_ident)
```
```   427     done
```
```   428 next
```
```   429   show "(\<lambda>h. norm (?inv (x + h) - ?inv x - - (?inv x * h * ?inv x)) / norm h)
```
```   430         -- 0 --> 0"
```
```   431   proof (rule LIM_equal2)
```
```   432     show "0 < norm x" using x by simp
```
```   433   next
```
```   434     fix h::'a
```
```   435     assume 1: "h \<noteq> 0"
```
```   436     assume "norm (h - 0) < norm x"
```
```   437     hence "h \<noteq> -x" by clarsimp
```
```   438     hence 2: "x + h \<noteq> 0"
```
```   439       apply (rule contrapos_nn)
```
```   440       apply (rule sym)
```
```   441       apply (erule minus_unique)
```
```   442       done
```
```   443     show "norm (?inv (x + h) - ?inv x - - (?inv x * h * ?inv x)) / norm h
```
```   444           = norm ((?inv (x + h) - ?inv x) * h * ?inv x) / norm h"
```
```   445       apply (subst inverse_diff_inverse [OF 2 x])
```
```   446       apply (subst minus_diff_minus)
```
```   447       apply (subst norm_minus_cancel)
```
```   448       apply (simp add: left_diff_distrib)
```
```   449       done
```
```   450   next
```
```   451     show "(\<lambda>h. norm ((?inv (x + h) - ?inv x) * h * ?inv x) / norm h)
```
```   452           -- 0 --> 0"
```
```   453     proof (rule real_LIM_sandwich_zero)
```
```   454       show "(\<lambda>h. norm (?inv (x + h) - ?inv x) * norm (?inv x))
```
```   455             -- 0 --> 0"
```
```   456         apply (rule LIM_mult_left_zero)
```
```   457         apply (rule LIM_norm_zero)
```
```   458         apply (rule LIM_zero)
```
```   459         apply (rule LIM_offset_zero)
```
```   460         apply (rule LIM_inverse)
```
```   461         apply (rule LIM_ident)
```
```   462         apply (rule x)
```
```   463         done
```
```   464     next
```
```   465       fix h::'a assume h: "h \<noteq> 0"
```
```   466       show "0 \<le> norm ((?inv (x + h) - ?inv x) * h * ?inv x) / norm h"
```
```   467         apply (rule divide_nonneg_pos)
```
```   468         apply (rule norm_ge_zero)
```
```   469         apply (simp add: h)
```
```   470         done
```
```   471     next
```
```   472       fix h::'a assume h: "h \<noteq> 0"
```
```   473       have "norm ((?inv (x + h) - ?inv x) * h * ?inv x) / norm h
```
```   474             \<le> norm (?inv (x + h) - ?inv x) * norm h * norm (?inv x) / norm h"
```
```   475         apply (rule divide_right_mono [OF _ norm_ge_zero])
```
```   476         apply (rule order_trans [OF norm_mult_ineq])
```
```   477         apply (rule mult_right_mono [OF _ norm_ge_zero])
```
```   478         apply (rule norm_mult_ineq)
```
```   479         done
```
```   480       also have "\<dots> = norm (?inv (x + h) - ?inv x) * norm (?inv x)"
```
```   481         by simp
```
```   482       finally show "norm ((?inv (x + h) - ?inv x) * h * ?inv x) / norm h
```
```   483             \<le> norm (?inv (x + h) - ?inv x) * norm (?inv x)" .
```
```   484     qed
```
```   485   qed
```
```   486 qed
```
```   487
```
```   488 subsection {* Alternate definition *}
```
```   489
```
```   490 lemma field_fderiv_def:
```
```   491   fixes x :: "'a::real_normed_field" shows
```
```   492   "FDERIV f x :> (\<lambda>h. h * D) = (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
```
```   493  apply (unfold fderiv_def)
```
```   494  apply (simp add: mult.bounded_linear_left)
```
```   495  apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
```
```   496  apply (subst diff_divide_distrib)
```
```   497  apply (subst times_divide_eq_left [symmetric])
```
```   498  apply (simp cong: LIM_cong)
```
```   499  apply (simp add: LIM_norm_zero_iff LIM_zero_iff)
```
```   500 done
```
```   501
```
```   502 end
```