src/HOL/Deriv.thy
 author hoelzl Tue Mar 26 12:21:00 2013 +0100 (2013-03-26) changeset 51529 2d2f59e6055a parent 51526 155263089e7b child 51641 cd05e9fcc63d permissions -rw-r--r--
move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
```     1 (*  Title       : Deriv.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
```
```     5     GMVT by Benjamin Porter, 2005
```
```     6 *)
```
```     7
```
```     8 header{* Differentiation *}
```
```     9
```
```    10 theory Deriv
```
```    11 imports Limits
```
```    12 begin
```
```    13
```
```    14 text{*Standard Definitions*}
```
```    15
```
```    16 definition
```
```    17   deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
```
```    18     --{*Differentiation: D is derivative of function f at x*}
```
```    19           ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
```
```    20   "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"
```
```    21
```
```    22 subsection {* Derivatives *}
```
```    23
```
```    24 lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
```
```    25 by (simp add: deriv_def)
```
```    26
```
```    27 lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
```
```    28 by (simp add: deriv_def)
```
```    29
```
```    30 lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
```
```    31   by (simp add: deriv_def tendsto_const)
```
```    32
```
```    33 lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
```
```    34   by (simp add: deriv_def tendsto_const cong: LIM_cong)
```
```    35
```
```    36 lemma DERIV_add:
```
```    37   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
```
```    38   by (simp only: deriv_def add_diff_add add_divide_distrib tendsto_add)
```
```    39
```
```    40 lemma DERIV_minus:
```
```    41   "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
```
```    42   by (simp only: deriv_def minus_diff_minus divide_minus_left tendsto_minus)
```
```    43
```
```    44 lemma DERIV_diff:
```
```    45   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
```
```    46 by (simp only: diff_minus DERIV_add DERIV_minus)
```
```    47
```
```    48 lemma DERIV_add_minus:
```
```    49   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
```
```    50 by (simp only: DERIV_add DERIV_minus)
```
```    51
```
```    52 lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
```
```    53 proof (unfold isCont_iff)
```
```    54   assume "DERIV f x :> D"
```
```    55   hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"
```
```    56     by (rule DERIV_D)
```
```    57   hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
```
```    58     by (intro tendsto_mult tendsto_ident_at)
```
```    59   hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
```
```    60     by simp
```
```    61   hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
```
```    62     by (simp cong: LIM_cong)
```
```    63   thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
```
```    64     by (simp add: LIM_def dist_norm)
```
```    65 qed
```
```    66
```
```    67 lemma DERIV_mult_lemma:
```
```    68   fixes a b c d :: "'a::real_field"
```
```    69   shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"
```
```    70   by (simp add: field_simps diff_divide_distrib)
```
```    71
```
```    72 lemma DERIV_mult':
```
```    73   assumes f: "DERIV f x :> D"
```
```    74   assumes g: "DERIV g x :> E"
```
```    75   shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
```
```    76 proof (unfold deriv_def)
```
```    77   from f have "isCont f x"
```
```    78     by (rule DERIV_isCont)
```
```    79   hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
```
```    80     by (simp only: isCont_iff)
```
```    81   hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +
```
```    82               ((f(x+h) - f x) / h) * g x)
```
```    83           -- 0 --> f x * E + D * g x"
```
```    84     by (intro tendsto_intros DERIV_D f g)
```
```    85   thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)
```
```    86          -- 0 --> f x * E + D * g x"
```
```    87     by (simp only: DERIV_mult_lemma)
```
```    88 qed
```
```    89
```
```    90 lemma DERIV_mult:
```
```    91     "DERIV f x :> Da \<Longrightarrow> DERIV g x :> Db \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x :> Da * g x + Db * f x"
```
```    92   by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
```
```    93
```
```    94 lemma DERIV_unique:
```
```    95     "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
```
```    96   unfolding deriv_def by (rule LIM_unique)
```
```    97
```
```    98 text{*Differentiation of finite sum*}
```
```    99
```
```   100 lemma DERIV_setsum:
```
```   101   assumes "finite S"
```
```   102   and "\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x :> (f' x n)"
```
```   103   shows "DERIV (%x. setsum (f x) S) x :> setsum (f' x) S"
```
```   104   using assms by induct (auto intro!: DERIV_add)
```
```   105
```
```   106 lemma DERIV_sumr [rule_format (no_asm)]:
```
```   107      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
```
```   108       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
```
```   109   by (auto intro: DERIV_setsum)
```
```   110
```
```   111 text{*Alternative definition for differentiability*}
```
```   112
```
```   113 lemma DERIV_LIM_iff:
```
```   114   fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
```
```   115      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
```
```   116       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
```
```   117 apply (rule iffI)
```
```   118 apply (drule_tac k="- a" in LIM_offset)
```
```   119 apply (simp add: diff_minus)
```
```   120 apply (drule_tac k="a" in LIM_offset)
```
```   121 apply (simp add: add_commute)
```
```   122 done
```
```   123
```
```   124 lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
```
```   125 by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
```
```   126
```
```   127 lemma DERIV_inverse_lemma:
```
```   128   "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
```
```   129    \<Longrightarrow> (inverse a - inverse b) / h
```
```   130      = - (inverse a * ((a - b) / h) * inverse b)"
```
```   131 by (simp add: inverse_diff_inverse)
```
```   132
```
```   133 lemma DERIV_inverse':
```
```   134   assumes der: "DERIV f x :> D"
```
```   135   assumes neq: "f x \<noteq> 0"
```
```   136   shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
```
```   137     (is "DERIV _ _ :> ?E")
```
```   138 proof (unfold DERIV_iff2)
```
```   139   from der have lim_f: "f -- x --> f x"
```
```   140     by (rule DERIV_isCont [unfolded isCont_def])
```
```   141
```
```   142   from neq have "0 < norm (f x)" by simp
```
```   143   with LIM_D [OF lim_f] obtain s
```
```   144     where s: "0 < s"
```
```   145     and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
```
```   146                   \<Longrightarrow> norm (f z - f x) < norm (f x)"
```
```   147     by fast
```
```   148
```
```   149   show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
```
```   150   proof (rule LIM_equal2 [OF s])
```
```   151     fix z
```
```   152     assume "z \<noteq> x" "norm (z - x) < s"
```
```   153     hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
```
```   154     hence "f z \<noteq> 0" by auto
```
```   155     thus "(inverse (f z) - inverse (f x)) / (z - x) =
```
```   156           - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"
```
```   157       using neq by (rule DERIV_inverse_lemma)
```
```   158   next
```
```   159     from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"
```
```   160       by (unfold DERIV_iff2)
```
```   161     thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
```
```   162           -- x --> ?E"
```
```   163       by (intro tendsto_intros lim_f neq)
```
```   164   qed
```
```   165 qed
```
```   166
```
```   167 lemma DERIV_divide:
```
```   168   "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
```
```   169    \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
```
```   170 apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
```
```   171           D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
```
```   172 apply (erule subst)
```
```   173 apply (unfold divide_inverse)
```
```   174 apply (erule DERIV_mult')
```
```   175 apply (erule (1) DERIV_inverse')
```
```   176 apply (simp add: ring_distribs nonzero_inverse_mult_distrib)
```
```   177 done
```
```   178
```
```   179 lemma DERIV_power_Suc:
```
```   180   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"
```
```   181   assumes f: "DERIV f x :> D"
```
```   182   shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"
```
```   183 proof (induct n)
```
```   184 case 0
```
```   185   show ?case by (simp add: f)
```
```   186 case (Suc k)
```
```   187   from DERIV_mult' [OF f Suc] show ?case
```
```   188     apply (simp only: of_nat_Suc ring_distribs mult_1_left)
```
```   189     apply (simp only: power_Suc algebra_simps)
```
```   190     done
```
```   191 qed
```
```   192
```
```   193 lemma DERIV_power:
```
```   194   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"
```
```   195   assumes f: "DERIV f x :> D"
```
```   196   shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
```
```   197 by (cases "n", simp, simp add: DERIV_power_Suc f del: power_Suc)
```
```   198
```
```   199 text {* Caratheodory formulation of derivative at a point *}
```
```   200
```
```   201 lemma CARAT_DERIV:
```
```   202      "(DERIV f x :> l) =
```
```   203       (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
```
```   204       (is "?lhs = ?rhs")
```
```   205 proof
```
```   206   assume der: "DERIV f x :> l"
```
```   207   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
```
```   208   proof (intro exI conjI)
```
```   209     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
```
```   210     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
```
```   211     show "isCont ?g x" using der
```
```   212       by (simp add: isCont_iff DERIV_iff diff_minus
```
```   213                cong: LIM_equal [rule_format])
```
```   214     show "?g x = l" by simp
```
```   215   qed
```
```   216 next
```
```   217   assume "?rhs"
```
```   218   then obtain g where
```
```   219     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
```
```   220   thus "(DERIV f x :> l)"
```
```   221      by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
```
```   222 qed
```
```   223
```
```   224 lemma DERIV_chain':
```
```   225   assumes f: "DERIV f x :> D"
```
```   226   assumes g: "DERIV g (f x) :> E"
```
```   227   shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
```
```   228 proof (unfold DERIV_iff2)
```
```   229   obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
```
```   230     and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
```
```   231     using CARAT_DERIV [THEN iffD1, OF g] by fast
```
```   232   from f have "f -- x --> f x"
```
```   233     by (rule DERIV_isCont [unfolded isCont_def])
```
```   234   with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
```
```   235     by (rule isCont_tendsto_compose)
```
```   236   hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
```
```   237           -- x --> d (f x) * D"
```
```   238     by (rule tendsto_mult [OF _ f [unfolded DERIV_iff2]])
```
```   239   thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
```
```   240     by (simp add: d dfx)
```
```   241 qed
```
```   242
```
```   243 text {*
```
```   244  Let's do the standard proof, though theorem
```
```   245  @{text "LIM_mult2"} follows from a NS proof
```
```   246 *}
```
```   247
```
```   248 lemma DERIV_cmult:
```
```   249       "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
```
```   250 by (drule DERIV_mult' [OF DERIV_const], simp)
```
```   251
```
```   252 lemma DERIV_cdivide: "DERIV f x :> D ==> DERIV (%x. f x / c) x :> D / c"
```
```   253   apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x :> (1 / c) * D", force)
```
```   254   apply (erule DERIV_cmult)
```
```   255   done
```
```   256
```
```   257 text {* Standard version *}
```
```   258 lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
```
```   259 by (drule (1) DERIV_chain', simp add: o_def mult_commute)
```
```   260
```
```   261 lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
```
```   262 by (auto dest: DERIV_chain simp add: o_def)
```
```   263
```
```   264 text {* Derivative of linear multiplication *}
```
```   265 lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
```
```   266 by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
```
```   267
```
```   268 lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
```
```   269 apply (cut_tac DERIV_power [OF DERIV_ident])
```
```   270 apply (simp add: real_of_nat_def)
```
```   271 done
```
```   272
```
```   273 text {* Power of @{text "-1"} *}
```
```   274
```
```   275 lemma DERIV_inverse:
```
```   276   fixes x :: "'a::{real_normed_field}"
```
```   277   shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
```
```   278 by (drule DERIV_inverse' [OF DERIV_ident]) simp
```
```   279
```
```   280 text {* Derivative of inverse *}
```
```   281 lemma DERIV_inverse_fun:
```
```   282   fixes x :: "'a::{real_normed_field}"
```
```   283   shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
```
```   284       ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
```
```   285 by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
```
```   286
```
```   287 text {* Derivative of quotient *}
```
```   288 lemma DERIV_quotient:
```
```   289   fixes x :: "'a::{real_normed_field}"
```
```   290   shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
```
```   291        ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
```
```   292 by (drule (2) DERIV_divide) (simp add: mult_commute)
```
```   293
```
```   294 text {* @{text "DERIV_intros"} *}
```
```   295 ML {*
```
```   296 structure Deriv_Intros = Named_Thms
```
```   297 (
```
```   298   val name = @{binding DERIV_intros}
```
```   299   val description = "DERIV introduction rules"
```
```   300 )
```
```   301 *}
```
```   302
```
```   303 setup Deriv_Intros.setup
```
```   304
```
```   305 lemma DERIV_cong: "\<lbrakk> DERIV f x :> X ; X = Y \<rbrakk> \<Longrightarrow> DERIV f x :> Y"
```
```   306   by simp
```
```   307
```
```   308 declare
```
```   309   DERIV_const[THEN DERIV_cong, DERIV_intros]
```
```   310   DERIV_ident[THEN DERIV_cong, DERIV_intros]
```
```   311   DERIV_add[THEN DERIV_cong, DERIV_intros]
```
```   312   DERIV_minus[THEN DERIV_cong, DERIV_intros]
```
```   313   DERIV_mult[THEN DERIV_cong, DERIV_intros]
```
```   314   DERIV_diff[THEN DERIV_cong, DERIV_intros]
```
```   315   DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
```
```   316   DERIV_divide[THEN DERIV_cong, DERIV_intros]
```
```   317   DERIV_power[where 'a=real, THEN DERIV_cong,
```
```   318               unfolded real_of_nat_def[symmetric], DERIV_intros]
```
```   319   DERIV_setsum[THEN DERIV_cong, DERIV_intros]
```
```   320
```
```   321
```
```   322 subsection {* Differentiability predicate *}
```
```   323
```
```   324 definition
```
```   325   differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
```
```   326     (infixl "differentiable" 60) where
```
```   327   "f differentiable x = (\<exists>D. DERIV f x :> D)"
```
```   328
```
```   329 lemma differentiableE [elim?]:
```
```   330   assumes "f differentiable x"
```
```   331   obtains df where "DERIV f x :> df"
```
```   332   using assms unfolding differentiable_def ..
```
```   333
```
```   334 lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
```
```   335 by (simp add: differentiable_def)
```
```   336
```
```   337 lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
```
```   338 by (force simp add: differentiable_def)
```
```   339
```
```   340 lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
```
```   341   by (rule DERIV_ident [THEN differentiableI])
```
```   342
```
```   343 lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
```
```   344   by (rule DERIV_const [THEN differentiableI])
```
```   345
```
```   346 lemma differentiable_compose:
```
```   347   assumes f: "f differentiable (g x)"
```
```   348   assumes g: "g differentiable x"
```
```   349   shows "(\<lambda>x. f (g x)) differentiable x"
```
```   350 proof -
```
```   351   from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
```
```   352   moreover
```
```   353   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   354   ultimately
```
```   355   have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
```
```   356   thus ?thesis by (rule differentiableI)
```
```   357 qed
```
```   358
```
```   359 lemma differentiable_sum [simp]:
```
```   360   assumes "f differentiable x"
```
```   361   and "g differentiable x"
```
```   362   shows "(\<lambda>x. f x + g x) differentiable x"
```
```   363 proof -
```
```   364   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   365   moreover
```
```   366   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   367   ultimately
```
```   368   have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
```
```   369   thus ?thesis by (rule differentiableI)
```
```   370 qed
```
```   371
```
```   372 lemma differentiable_minus [simp]:
```
```   373   assumes "f differentiable x"
```
```   374   shows "(\<lambda>x. - f x) differentiable x"
```
```   375 proof -
```
```   376   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   377   hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
```
```   378   thus ?thesis by (rule differentiableI)
```
```   379 qed
```
```   380
```
```   381 lemma differentiable_diff [simp]:
```
```   382   assumes "f differentiable x"
```
```   383   assumes "g differentiable x"
```
```   384   shows "(\<lambda>x. f x - g x) differentiable x"
```
```   385   unfolding diff_minus using assms by simp
```
```   386
```
```   387 lemma differentiable_mult [simp]:
```
```   388   assumes "f differentiable x"
```
```   389   assumes "g differentiable x"
```
```   390   shows "(\<lambda>x. f x * g x) differentiable x"
```
```   391 proof -
```
```   392   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   393   moreover
```
```   394   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   395   ultimately
```
```   396   have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
```
```   397   thus ?thesis by (rule differentiableI)
```
```   398 qed
```
```   399
```
```   400 lemma differentiable_inverse [simp]:
```
```   401   assumes "f differentiable x" and "f x \<noteq> 0"
```
```   402   shows "(\<lambda>x. inverse (f x)) differentiable x"
```
```   403 proof -
```
```   404   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   405   hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
```
```   406     using `f x \<noteq> 0` by (rule DERIV_inverse')
```
```   407   thus ?thesis by (rule differentiableI)
```
```   408 qed
```
```   409
```
```   410 lemma differentiable_divide [simp]:
```
```   411   assumes "f differentiable x"
```
```   412   assumes "g differentiable x" and "g x \<noteq> 0"
```
```   413   shows "(\<lambda>x. f x / g x) differentiable x"
```
```   414   unfolding divide_inverse using assms by simp
```
```   415
```
```   416 lemma differentiable_power [simp]:
```
```   417   fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a"
```
```   418   assumes "f differentiable x"
```
```   419   shows "(\<lambda>x. f x ^ n) differentiable x"
```
```   420   apply (induct n)
```
```   421   apply simp
```
```   422   apply (simp add: assms)
```
```   423   done
```
```   424
```
```   425 subsection {* Local extrema *}
```
```   426
```
```   427 text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
```
```   428
```
```   429 lemma DERIV_pos_inc_right:
```
```   430   fixes f :: "real => real"
```
```   431   assumes der: "DERIV f x :> l"
```
```   432       and l:   "0 < l"
```
```   433   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
```
```   434 proof -
```
```   435   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
```
```   436   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)"
```
```   437     by (simp add: diff_minus)
```
```   438   then obtain s
```
```   439         where s:   "0 < s"
```
```   440           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
```
```   441     by auto
```
```   442   thus ?thesis
```
```   443   proof (intro exI conjI strip)
```
```   444     show "0<s" using s .
```
```   445     fix h::real
```
```   446     assume "0 < h" "h < s"
```
```   447     with all [of h] show "f x < f (x+h)"
```
```   448     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   449     split add: split_if_asm)
```
```   450       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
```
```   451       with l
```
```   452       have "0 < (f (x+h) - f x) / h" by arith
```
```   453       thus "f x < f (x+h)"
```
```   454   by (simp add: pos_less_divide_eq h)
```
```   455     qed
```
```   456   qed
```
```   457 qed
```
```   458
```
```   459 lemma DERIV_neg_dec_left:
```
```   460   fixes f :: "real => real"
```
```   461   assumes der: "DERIV f x :> l"
```
```   462       and l:   "l < 0"
```
```   463   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
```
```   464 proof -
```
```   465   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
```
```   466   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)"
```
```   467     by (simp add: diff_minus)
```
```   468   then obtain s
```
```   469         where s:   "0 < s"
```
```   470           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
```
```   471     by auto
```
```   472   thus ?thesis
```
```   473   proof (intro exI conjI strip)
```
```   474     show "0<s" using s .
```
```   475     fix h::real
```
```   476     assume "0 < h" "h < s"
```
```   477     with all [of "-h"] show "f x < f (x-h)"
```
```   478     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   479     split add: split_if_asm)
```
```   480       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
```
```   481       with l
```
```   482       have "0 < (f (x-h) - f x) / h" by arith
```
```   483       thus "f x < f (x-h)"
```
```   484   by (simp add: pos_less_divide_eq h)
```
```   485     qed
```
```   486   qed
```
```   487 qed
```
```   488
```
```   489 lemma DERIV_pos_inc_left:
```
```   490   fixes f :: "real => real"
```
```   491   shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
```
```   492   apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])
```
```   493   apply (auto simp add: DERIV_minus)
```
```   494   done
```
```   495
```
```   496 lemma DERIV_neg_dec_right:
```
```   497   fixes f :: "real => real"
```
```   498   shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
```
```   499   apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])
```
```   500   apply (auto simp add: DERIV_minus)
```
```   501   done
```
```   502
```
```   503 lemma DERIV_local_max:
```
```   504   fixes f :: "real => real"
```
```   505   assumes der: "DERIV f x :> l"
```
```   506       and d:   "0 < d"
```
```   507       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
```
```   508   shows "l = 0"
```
```   509 proof (cases rule: linorder_cases [of l 0])
```
```   510   case equal thus ?thesis .
```
```   511 next
```
```   512   case less
```
```   513   from DERIV_neg_dec_left [OF der less]
```
```   514   obtain d' where d': "0 < d'"
```
```   515              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
```
```   516   from real_lbound_gt_zero [OF d d']
```
```   517   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   518   with lt le [THEN spec [where x="x-e"]]
```
```   519   show ?thesis by (auto simp add: abs_if)
```
```   520 next
```
```   521   case greater
```
```   522   from DERIV_pos_inc_right [OF der greater]
```
```   523   obtain d' where d': "0 < d'"
```
```   524              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
```
```   525   from real_lbound_gt_zero [OF d d']
```
```   526   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   527   with lt le [THEN spec [where x="x+e"]]
```
```   528   show ?thesis by (auto simp add: abs_if)
```
```   529 qed
```
```   530
```
```   531
```
```   532 text{*Similar theorem for a local minimum*}
```
```   533 lemma DERIV_local_min:
```
```   534   fixes f :: "real => real"
```
```   535   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
```
```   536 by (drule DERIV_minus [THEN DERIV_local_max], auto)
```
```   537
```
```   538
```
```   539 text{*In particular, if a function is locally flat*}
```
```   540 lemma DERIV_local_const:
```
```   541   fixes f :: "real => real"
```
```   542   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
```
```   543 by (auto dest!: DERIV_local_max)
```
```   544
```
```   545
```
```   546 subsection {* Rolle's Theorem *}
```
```   547
```
```   548 text{*Lemma about introducing open ball in open interval*}
```
```   549 lemma lemma_interval_lt:
```
```   550      "[| a < x;  x < b |]
```
```   551       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
```
```   552
```
```   553 apply (simp add: abs_less_iff)
```
```   554 apply (insert linorder_linear [of "x-a" "b-x"], safe)
```
```   555 apply (rule_tac x = "x-a" in exI)
```
```   556 apply (rule_tac [2] x = "b-x" in exI, auto)
```
```   557 done
```
```   558
```
```   559 lemma lemma_interval: "[| a < x;  x < b |] ==>
```
```   560         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
```
```   561 apply (drule lemma_interval_lt, auto)
```
```   562 apply force
```
```   563 done
```
```   564
```
```   565 text{*Rolle's Theorem.
```
```   566    If @{term f} is defined and continuous on the closed interval
```
```   567    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
```
```   568    and @{term "f(a) = f(b)"},
```
```   569    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
```
```   570 theorem Rolle:
```
```   571   assumes lt: "a < b"
```
```   572       and eq: "f(a) = f(b)"
```
```   573       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```   574       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```   575   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
```
```   576 proof -
```
```   577   have le: "a \<le> b" using lt by simp
```
```   578   from isCont_eq_Ub [OF le con]
```
```   579   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
```
```   580              and alex: "a \<le> x" and xleb: "x \<le> b"
```
```   581     by blast
```
```   582   from isCont_eq_Lb [OF le con]
```
```   583   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
```
```   584               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
```
```   585     by blast
```
```   586   show ?thesis
```
```   587   proof cases
```
```   588     assume axb: "a < x & x < b"
```
```   589         --{*@{term f} attains its maximum within the interval*}
```
```   590     hence ax: "a<x" and xb: "x<b" by arith +
```
```   591     from lemma_interval [OF ax xb]
```
```   592     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   593       by blast
```
```   594     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
```
```   595       by blast
```
```   596     from differentiableD [OF dif [OF axb]]
```
```   597     obtain l where der: "DERIV f x :> l" ..
```
```   598     have "l=0" by (rule DERIV_local_max [OF der d bound'])
```
```   599         --{*the derivative at a local maximum is zero*}
```
```   600     thus ?thesis using ax xb der by auto
```
```   601   next
```
```   602     assume notaxb: "~ (a < x & x < b)"
```
```   603     hence xeqab: "x=a | x=b" using alex xleb by arith
```
```   604     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
```
```   605     show ?thesis
```
```   606     proof cases
```
```   607       assume ax'b: "a < x' & x' < b"
```
```   608         --{*@{term f} attains its minimum within the interval*}
```
```   609       hence ax': "a<x'" and x'b: "x'<b" by arith+
```
```   610       from lemma_interval [OF ax' x'b]
```
```   611       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   612   by blast
```
```   613       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
```
```   614   by blast
```
```   615       from differentiableD [OF dif [OF ax'b]]
```
```   616       obtain l where der: "DERIV f x' :> l" ..
```
```   617       have "l=0" by (rule DERIV_local_min [OF der d bound'])
```
```   618         --{*the derivative at a local minimum is zero*}
```
```   619       thus ?thesis using ax' x'b der by auto
```
```   620     next
```
```   621       assume notax'b: "~ (a < x' & x' < b)"
```
```   622         --{*@{term f} is constant througout the interval*}
```
```   623       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
```
```   624       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
```
```   625       from dense [OF lt]
```
```   626       obtain r where ar: "a < r" and rb: "r < b" by blast
```
```   627       from lemma_interval [OF ar rb]
```
```   628       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   629   by blast
```
```   630       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
```
```   631       proof (clarify)
```
```   632         fix z::real
```
```   633         assume az: "a \<le> z" and zb: "z \<le> b"
```
```   634         show "f z = f b"
```
```   635         proof (rule order_antisym)
```
```   636           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
```
```   637           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
```
```   638         qed
```
```   639       qed
```
```   640       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
```
```   641       proof (intro strip)
```
```   642         fix y::real
```
```   643         assume lt: "\<bar>r-y\<bar> < d"
```
```   644         hence "f y = f b" by (simp add: eq_fb bound)
```
```   645         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
```
```   646       qed
```
```   647       from differentiableD [OF dif [OF conjI [OF ar rb]]]
```
```   648       obtain l where der: "DERIV f r :> l" ..
```
```   649       have "l=0" by (rule DERIV_local_const [OF der d bound'])
```
```   650         --{*the derivative of a constant function is zero*}
```
```   651       thus ?thesis using ar rb der by auto
```
```   652     qed
```
```   653   qed
```
```   654 qed
```
```   655
```
```   656
```
```   657 subsection{*Mean Value Theorem*}
```
```   658
```
```   659 lemma lemma_MVT:
```
```   660      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
```
```   661   by (cases "a = b") (simp_all add: field_simps)
```
```   662
```
```   663 theorem MVT:
```
```   664   assumes lt:  "a < b"
```
```   665       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```   666       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```   667   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
```
```   668                    (f(b) - f(a) = (b-a) * l)"
```
```   669 proof -
```
```   670   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
```
```   671   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
```
```   672     using con by (fast intro: isCont_intros)
```
```   673   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
```
```   674   proof (clarify)
```
```   675     fix x::real
```
```   676     assume ax: "a < x" and xb: "x < b"
```
```   677     from differentiableD [OF dif [OF conjI [OF ax xb]]]
```
```   678     obtain l where der: "DERIV f x :> l" ..
```
```   679     show "?F differentiable x"
```
```   680       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
```
```   681           blast intro: DERIV_diff DERIV_cmult_Id der)
```
```   682   qed
```
```   683   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
```
```   684   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
```
```   685     by blast
```
```   686   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
```
```   687     by (rule DERIV_cmult_Id)
```
```   688   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
```
```   689                    :> 0 + (f b - f a) / (b - a)"
```
```   690     by (rule DERIV_add [OF der])
```
```   691   show ?thesis
```
```   692   proof (intro exI conjI)
```
```   693     show "a < z" using az .
```
```   694     show "z < b" using zb .
```
```   695     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
```
```   696     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
```
```   697   qed
```
```   698 qed
```
```   699
```
```   700 lemma MVT2:
```
```   701      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
```
```   702       ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
```
```   703 apply (drule MVT)
```
```   704 apply (blast intro: DERIV_isCont)
```
```   705 apply (force dest: order_less_imp_le simp add: differentiable_def)
```
```   706 apply (blast dest: DERIV_unique order_less_imp_le)
```
```   707 done
```
```   708
```
```   709
```
```   710 text{*A function is constant if its derivative is 0 over an interval.*}
```
```   711
```
```   712 lemma DERIV_isconst_end:
```
```   713   fixes f :: "real => real"
```
```   714   shows "[| a < b;
```
```   715          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```   716          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```   717         ==> f b = f a"
```
```   718 apply (drule MVT, assumption)
```
```   719 apply (blast intro: differentiableI)
```
```   720 apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
```
```   721 done
```
```   722
```
```   723 lemma DERIV_isconst1:
```
```   724   fixes f :: "real => real"
```
```   725   shows "[| a < b;
```
```   726          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```   727          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```   728         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
```
```   729 apply safe
```
```   730 apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
```
```   731 apply (drule_tac b = x in DERIV_isconst_end, auto)
```
```   732 done
```
```   733
```
```   734 lemma DERIV_isconst2:
```
```   735   fixes f :: "real => real"
```
```   736   shows "[| a < b;
```
```   737          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```   738          \<forall>x. a < x & x < b --> DERIV f x :> 0;
```
```   739          a \<le> x; x \<le> b |]
```
```   740         ==> f x = f a"
```
```   741 apply (blast dest: DERIV_isconst1)
```
```   742 done
```
```   743
```
```   744 lemma DERIV_isconst3: fixes a b x y :: real
```
```   745   assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
```
```   746   assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
```
```   747   shows "f x = f y"
```
```   748 proof (cases "x = y")
```
```   749   case False
```
```   750   let ?a = "min x y"
```
```   751   let ?b = "max x y"
```
```   752
```
```   753   have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
```
```   754   proof (rule allI, rule impI)
```
```   755     fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
```
```   756     hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
```
```   757     hence "z \<in> {a<..<b}" by auto
```
```   758     thus "DERIV f z :> 0" by (rule derivable)
```
```   759   qed
```
```   760   hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
```
```   761     and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
```
```   762
```
```   763   have "?a < ?b" using `x \<noteq> y` by auto
```
```   764   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
```
```   765   show ?thesis by auto
```
```   766 qed auto
```
```   767
```
```   768 lemma DERIV_isconst_all:
```
```   769   fixes f :: "real => real"
```
```   770   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
```
```   771 apply (rule linorder_cases [of x y])
```
```   772 apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
```
```   773 done
```
```   774
```
```   775 lemma DERIV_const_ratio_const:
```
```   776   fixes f :: "real => real"
```
```   777   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
```
```   778 apply (rule linorder_cases [of a b], auto)
```
```   779 apply (drule_tac [!] f = f in MVT)
```
```   780 apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
```
```   781 apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
```
```   782 done
```
```   783
```
```   784 lemma DERIV_const_ratio_const2:
```
```   785   fixes f :: "real => real"
```
```   786   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
```
```   787 apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
```
```   788 apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
```
```   789 done
```
```   790
```
```   791 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
```
```   792 by (simp)
```
```   793
```
```   794 lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
```
```   795 by (simp)
```
```   796
```
```   797 text{*Gallileo's "trick": average velocity = av. of end velocities*}
```
```   798
```
```   799 lemma DERIV_const_average:
```
```   800   fixes v :: "real => real"
```
```   801   assumes neq: "a \<noteq> (b::real)"
```
```   802       and der: "\<forall>x. DERIV v x :> k"
```
```   803   shows "v ((a + b)/2) = (v a + v b)/2"
```
```   804 proof (cases rule: linorder_cases [of a b])
```
```   805   case equal with neq show ?thesis by simp
```
```   806 next
```
```   807   case less
```
```   808   have "(v b - v a) / (b - a) = k"
```
```   809     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```   810   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```   811   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
```
```   812     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```   813   ultimately show ?thesis using neq by force
```
```   814 next
```
```   815   case greater
```
```   816   have "(v b - v a) / (b - a) = k"
```
```   817     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```   818   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```   819   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
```
```   820     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```   821   ultimately show ?thesis using neq by (force simp add: add_commute)
```
```   822 qed
```
```   823
```
```   824 (* A function with positive derivative is increasing.
```
```   825    A simple proof using the MVT, by Jeremy Avigad. And variants.
```
```   826 *)
```
```   827 lemma DERIV_pos_imp_increasing:
```
```   828   fixes a::real and b::real and f::"real => real"
```
```   829   assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
```
```   830   shows "f a < f b"
```
```   831 proof (rule ccontr)
```
```   832   assume f: "~ f a < f b"
```
```   833   have "EX l z. a < z & z < b & DERIV f z :> l
```
```   834       & f b - f a = (b - a) * l"
```
```   835     apply (rule MVT)
```
```   836       using assms
```
```   837       apply auto
```
```   838       apply (metis DERIV_isCont)
```
```   839      apply (metis differentiableI less_le)
```
```   840     done
```
```   841   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
```
```   842       and "f b - f a = (b - a) * l"
```
```   843     by auto
```
```   844   with assms f have "~(l > 0)"
```
```   845     by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
```
```   846   with assms z show False
```
```   847     by (metis DERIV_unique less_le)
```
```   848 qed
```
```   849
```
```   850 lemma DERIV_nonneg_imp_nondecreasing:
```
```   851   fixes a::real and b::real and f::"real => real"
```
```   852   assumes "a \<le> b" and
```
```   853     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
```
```   854   shows "f a \<le> f b"
```
```   855 proof (rule ccontr, cases "a = b")
```
```   856   assume "~ f a \<le> f b" and "a = b"
```
```   857   then show False by auto
```
```   858 next
```
```   859   assume A: "~ f a \<le> f b"
```
```   860   assume B: "a ~= b"
```
```   861   with assms have "EX l z. a < z & z < b & DERIV f z :> l
```
```   862       & f b - f a = (b - a) * l"
```
```   863     apply -
```
```   864     apply (rule MVT)
```
```   865       apply auto
```
```   866       apply (metis DERIV_isCont)
```
```   867      apply (metis differentiableI less_le)
```
```   868     done
```
```   869   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
```
```   870       and C: "f b - f a = (b - a) * l"
```
```   871     by auto
```
```   872   with A have "a < b" "f b < f a" by auto
```
```   873   with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
```
```   874     (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
```
```   875   with assms z show False
```
```   876     by (metis DERIV_unique order_less_imp_le)
```
```   877 qed
```
```   878
```
```   879 lemma DERIV_neg_imp_decreasing:
```
```   880   fixes a::real and b::real and f::"real => real"
```
```   881   assumes "a < b" and
```
```   882     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
```
```   883   shows "f a > f b"
```
```   884 proof -
```
```   885   have "(%x. -f x) a < (%x. -f x) b"
```
```   886     apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
```
```   887     using assms
```
```   888     apply auto
```
```   889     apply (metis DERIV_minus neg_0_less_iff_less)
```
```   890     done
```
```   891   thus ?thesis
```
```   892     by simp
```
```   893 qed
```
```   894
```
```   895 lemma DERIV_nonpos_imp_nonincreasing:
```
```   896   fixes a::real and b::real and f::"real => real"
```
```   897   assumes "a \<le> b" and
```
```   898     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
```
```   899   shows "f a \<ge> f b"
```
```   900 proof -
```
```   901   have "(%x. -f x) a \<le> (%x. -f x) b"
```
```   902     apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
```
```   903     using assms
```
```   904     apply auto
```
```   905     apply (metis DERIV_minus neg_0_le_iff_le)
```
```   906     done
```
```   907   thus ?thesis
```
```   908     by simp
```
```   909 qed
```
```   910
```
```   911 text {* Derivative of inverse function *}
```
```   912
```
```   913 lemma DERIV_inverse_function:
```
```   914   fixes f g :: "real \<Rightarrow> real"
```
```   915   assumes der: "DERIV f (g x) :> D"
```
```   916   assumes neq: "D \<noteq> 0"
```
```   917   assumes a: "a < x" and b: "x < b"
```
```   918   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
```
```   919   assumes cont: "isCont g x"
```
```   920   shows "DERIV g x :> inverse D"
```
```   921 unfolding DERIV_iff2
```
```   922 proof (rule LIM_equal2)
```
```   923   show "0 < min (x - a) (b - x)"
```
```   924     using a b by arith
```
```   925 next
```
```   926   fix y
```
```   927   assume "norm (y - x) < min (x - a) (b - x)"
```
```   928   hence "a < y" and "y < b"
```
```   929     by (simp_all add: abs_less_iff)
```
```   930   thus "(g y - g x) / (y - x) =
```
```   931         inverse ((f (g y) - x) / (g y - g x))"
```
```   932     by (simp add: inj)
```
```   933 next
```
```   934   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
```
```   935     by (rule der [unfolded DERIV_iff2])
```
```   936   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
```
```   937     using inj a b by simp
```
```   938   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
```
```   939   proof (safe intro!: exI)
```
```   940     show "0 < min (x - a) (b - x)"
```
```   941       using a b by simp
```
```   942   next
```
```   943     fix y
```
```   944     assume "norm (y - x) < min (x - a) (b - x)"
```
```   945     hence y: "a < y" "y < b"
```
```   946       by (simp_all add: abs_less_iff)
```
```   947     assume "g y = g x"
```
```   948     hence "f (g y) = f (g x)" by simp
```
```   949     hence "y = x" using inj y a b by simp
```
```   950     also assume "y \<noteq> x"
```
```   951     finally show False by simp
```
```   952   qed
```
```   953   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
```
```   954     using cont 1 2 by (rule isCont_LIM_compose2)
```
```   955   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
```
```   956         -- x --> inverse D"
```
```   957     using neq by (rule tendsto_inverse)
```
```   958 qed
```
```   959
```
```   960 subsection {* Generalized Mean Value Theorem *}
```
```   961
```
```   962 theorem GMVT:
```
```   963   fixes a b :: real
```
```   964   assumes alb: "a < b"
```
```   965     and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```   966     and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
```
```   967     and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
```
```   968     and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
```
```   969   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)"
```
```   970 proof -
```
```   971   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
```
```   972   from assms have "a < b" by simp
```
```   973   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
```
```   974     using fc gc by simp
```
```   975   moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
```
```   976     using fd gd by simp
```
```   977   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)
```
```   978   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" ..
```
```   979   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
```
```   980
```
```   981   from cdef have cint: "a < c \<and> c < b" by auto
```
```   982   with gd have "g differentiable c" by simp
```
```   983   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
```
```   984   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
```
```   985
```
```   986   from cdef have "a < c \<and> c < b" by auto
```
```   987   with fd have "f differentiable c" by simp
```
```   988   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
```
```   989   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
```
```   990
```
```   991   from cdef have "DERIV ?h c :> l" by auto
```
```   992   moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
```
```   993     using g'cdef f'cdef by (auto intro!: DERIV_intros)
```
```   994   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
```
```   995
```
```   996   {
```
```   997     from cdef have "?h b - ?h a = (b - a) * l" by auto
```
```   998     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```   999     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1000   }
```
```  1001   moreover
```
```  1002   {
```
```  1003     have "?h b - ?h a =
```
```  1004          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
```
```  1005           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
```
```  1006       by (simp add: algebra_simps)
```
```  1007     hence "?h b - ?h a = 0" by auto
```
```  1008   }
```
```  1009   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
```
```  1010   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
```
```  1011   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
```
```  1012   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
```
```  1013
```
```  1014   with g'cdef f'cdef cint show ?thesis by auto
```
```  1015 qed
```
```  1016
```
```  1017 lemma GMVT':
```
```  1018   fixes f g :: "real \<Rightarrow> real"
```
```  1019   assumes "a < b"
```
```  1020   assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
```
```  1021   assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
```
```  1022   assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
```
```  1023   assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
```
```  1024   shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
```
```  1025 proof -
```
```  1026   have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
```
```  1027     a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
```
```  1028     using assms by (intro GMVT) (force simp: differentiable_def)+
```
```  1029   then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
```
```  1030     using DERIV_f DERIV_g by (force dest: DERIV_unique)
```
```  1031   then show ?thesis
```
```  1032     by auto
```
```  1033 qed
```
```  1034
```
```  1035
```
```  1036 subsection {* L'Hopitals rule *}
```
```  1037
```
```  1038 lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
```
```  1039     DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
```
```  1040   unfolding DERIV_iff2
```
```  1041 proof (rule filterlim_cong)
```
```  1042   assume "eventually (\<lambda>x. f x = g x) (nhds x)"
```
```  1043   moreover then have "f x = g x" by (auto simp: eventually_nhds)
```
```  1044   moreover assume "x = y" "u = v"
```
```  1045   ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
```
```  1046     by (auto simp: eventually_within at_def elim: eventually_elim1)
```
```  1047 qed simp_all
```
```  1048
```
```  1049 lemma DERIV_shift:
```
```  1050   "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
```
```  1051   by (simp add: DERIV_iff field_simps)
```
```  1052
```
```  1053 lemma DERIV_mirror:
```
```  1054   "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
```
```  1055   by (simp add: deriv_def filterlim_at_split filterlim_at_left_to_right
```
```  1056                 tendsto_minus_cancel_left field_simps conj_commute)
```
```  1057
```
```  1058 lemma lhopital_right_0:
```
```  1059   fixes f0 g0 :: "real \<Rightarrow> real"
```
```  1060   assumes f_0: "(f0 ---> 0) (at_right 0)"
```
```  1061   assumes g_0: "(g0 ---> 0) (at_right 0)"
```
```  1062   assumes ev:
```
```  1063     "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
```
```  1064     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
```
```  1065     "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
```
```  1066     "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
```
```  1067   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
```
```  1068   shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
```
```  1069 proof -
```
```  1070   def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
```
```  1071   then have "f 0 = 0" by simp
```
```  1072
```
```  1073   def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
```
```  1074   then have "g 0 = 0" by simp
```
```  1075
```
```  1076   have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
```
```  1077       DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
```
```  1078     using ev by eventually_elim auto
```
```  1079   then obtain a where [arith]: "0 < a"
```
```  1080     and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
```
```  1081     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
```
```  1082     and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
```
```  1083     and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
```
```  1084     unfolding eventually_within eventually_at by (auto simp: dist_real_def)
```
```  1085
```
```  1086   have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
```
```  1087     using g0_neq_0 by (simp add: g_def)
```
```  1088
```
```  1089   { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
```
```  1090       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
```
```  1091          (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
```
```  1092   note f = this
```
```  1093
```
```  1094   { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
```
```  1095       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
```
```  1096          (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
```
```  1097   note g = this
```
```  1098
```
```  1099   have "isCont f 0"
```
```  1100     using tendsto_const[of "0::real" "at 0"] f_0
```
```  1101     unfolding isCont_def f_def
```
```  1102     by (intro filterlim_split_at_real)
```
```  1103        (auto elim: eventually_elim1
```
```  1104              simp add: filterlim_def le_filter_def eventually_within eventually_filtermap)
```
```  1105
```
```  1106   have "isCont g 0"
```
```  1107     using tendsto_const[of "0::real" "at 0"] g_0
```
```  1108     unfolding isCont_def g_def
```
```  1109     by (intro filterlim_split_at_real)
```
```  1110        (auto elim: eventually_elim1
```
```  1111              simp add: filterlim_def le_filter_def eventually_within eventually_filtermap)
```
```  1112
```
```  1113   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)"
```
```  1114   proof (rule bchoice, rule)
```
```  1115     fix x assume "x \<in> {0 <..< a}"
```
```  1116     then have x[arith]: "0 < x" "x < a" by auto
```
```  1117     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"
```
```  1118       by auto
```
```  1119     have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
```
```  1120       using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
```
```  1121     moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
```
```  1122       using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
```
```  1123     ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
```
```  1124       using f g `x < a` by (intro GMVT') auto
```
```  1125     then guess c ..
```
```  1126     moreover
```
```  1127     with g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
```
```  1128       by (simp add: field_simps)
```
```  1129     ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
```
```  1130       using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
```
```  1131   qed
```
```  1132   then guess \<zeta> ..
```
```  1133   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)"
```
```  1134     unfolding eventually_within eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
```
```  1135   moreover
```
```  1136   from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
```
```  1137     by eventually_elim auto
```
```  1138   then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
```
```  1139     by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
```
```  1140        (auto intro: tendsto_const tendsto_ident_at_within)
```
```  1141   then have "(\<zeta> ---> 0) (at_right 0)"
```
```  1142     by (rule tendsto_norm_zero_cancel)
```
```  1143   with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
```
```  1144     by (auto elim!: eventually_elim1 simp: filterlim_within filterlim_at)
```
```  1145   from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
```
```  1146     by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
```
```  1147   ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
```
```  1148     by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
```
```  1149        (auto elim: eventually_elim1)
```
```  1150   also have "?P \<longleftrightarrow> ?thesis"
```
```  1151     by (rule filterlim_cong) (auto simp: f_def g_def eventually_within)
```
```  1152   finally show ?thesis .
```
```  1153 qed
```
```  1154
```
```  1155 lemma lhopital_right:
```
```  1156   "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
```
```  1157     eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1158     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1159     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
```
```  1160     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
```
```  1161     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
```
```  1162   ((\<lambda> x. f x / g x) ---> y) (at_right x)"
```
```  1163   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
```
```  1164   by (rule lhopital_right_0)
```
```  1165
```
```  1166 lemma lhopital_left:
```
```  1167   "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
```
```  1168     eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1169     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1170     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
```
```  1171     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
```
```  1172     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
```
```  1173   ((\<lambda> x. f x / g x) ---> y) (at_left x)"
```
```  1174   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
```
```  1175   by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
```
```  1176
```
```  1177 lemma lhopital:
```
```  1178   "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
```
```  1179     eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1180     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1181     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
```
```  1182     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
```
```  1183     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
```
```  1184   ((\<lambda> x. f x / g x) ---> y) (at x)"
```
```  1185   unfolding eventually_at_split filterlim_at_split
```
```  1186   by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
```
```  1187
```
```  1188 lemma lhopital_right_0_at_top:
```
```  1189   fixes f g :: "real \<Rightarrow> real"
```
```  1190   assumes g_0: "LIM x at_right 0. g x :> at_top"
```
```  1191   assumes ev:
```
```  1192     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
```
```  1193     "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
```
```  1194     "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
```
```  1195   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
```
```  1196   shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
```
```  1197   unfolding tendsto_iff
```
```  1198 proof safe
```
```  1199   fix e :: real assume "0 < e"
```
```  1200
```
```  1201   with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
```
```  1202   have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
```
```  1203   from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
```
```  1204   obtain a where [arith]: "0 < a"
```
```  1205     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
```
```  1206     and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
```
```  1207     and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
```
```  1208     and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
```
```  1209     unfolding eventually_within_le by (auto simp: dist_real_def)
```
```  1210
```
```  1211   from Df have
```
```  1212     "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
```
```  1213     unfolding eventually_within eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
```
```  1214
```
```  1215   moreover
```
```  1216   have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
```
```  1217     using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
```
```  1218
```
```  1219   moreover
```
```  1220   have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
```
```  1221     using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
```
```  1222     by (rule filterlim_compose)
```
```  1223   then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
```
```  1224     by (intro tendsto_intros)
```
```  1225   then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
```
```  1226     by (simp add: inverse_eq_divide)
```
```  1227   from this[unfolded tendsto_iff, rule_format, of 1]
```
```  1228   have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
```
```  1229     by (auto elim!: eventually_elim1 simp: dist_real_def)
```
```  1230
```
```  1231   moreover
```
```  1232   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)"
```
```  1233     by (intro tendsto_intros)
```
```  1234   then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
```
```  1235     by (simp add: inverse_eq_divide)
```
```  1236   from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
```
```  1237   have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
```
```  1238     by (auto simp: dist_real_def)
```
```  1239
```
```  1240   ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
```
```  1241   proof eventually_elim
```
```  1242     fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
```
```  1243     assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
```
```  1244
```
```  1245     have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
```
```  1246       using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
```
```  1247     then guess y ..
```
```  1248     from this
```
```  1249     have [arith]: "t < y" "y < a" and D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
```
```  1250       using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
```
```  1251
```
```  1252     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"
```
```  1253       by (simp add: field_simps)
```
```  1254     have "norm (f t / g t - x) \<le>
```
```  1255         norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
```
```  1256       unfolding * by (rule norm_triangle_ineq)
```
```  1257     also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
```
```  1258       by (simp add: abs_mult D_eq dist_real_def)
```
```  1259     also have "\<dots> < (e / 4) * 2 + e / 2"
```
```  1260       using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
```
```  1261     finally show "dist (f t / g t) x < e"
```
```  1262       by (simp add: dist_real_def)
```
```  1263   qed
```
```  1264 qed
```
```  1265
```
```  1266 lemma lhopital_right_at_top:
```
```  1267   "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1268     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1269     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
```
```  1270     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
```
```  1271     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
```
```  1272     ((\<lambda> x. f x / g x) ---> y) (at_right x)"
```
```  1273   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
```
```  1274   by (rule lhopital_right_0_at_top)
```
```  1275
```
```  1276 lemma lhopital_left_at_top:
```
```  1277   "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1278     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1279     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
```
```  1280     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
```
```  1281     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
```
```  1282     ((\<lambda> x. f x / g x) ---> y) (at_left x)"
```
```  1283   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
```
```  1284   by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
```
```  1285
```
```  1286 lemma lhopital_at_top:
```
```  1287   "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1288     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1289     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
```
```  1290     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
```
```  1291     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
```
```  1292     ((\<lambda> x. f x / g x) ---> y) (at x)"
```
```  1293   unfolding eventually_at_split filterlim_at_split
```
```  1294   by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
```
```  1295
```
```  1296 lemma lhospital_at_top_at_top:
```
```  1297   fixes f g :: "real \<Rightarrow> real"
```
```  1298   assumes g_0: "LIM x at_top. g x :> at_top"
```
```  1299   assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
```
```  1300   assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
```
```  1301   assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
```
```  1302   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
```
```  1303   shows "((\<lambda> x. f x / g x) ---> x) at_top"
```
```  1304   unfolding filterlim_at_top_to_right
```
```  1305 proof (rule lhopital_right_0_at_top)
```
```  1306   let ?F = "\<lambda>x. f (inverse x)"
```
```  1307   let ?G = "\<lambda>x. g (inverse x)"
```
```  1308   let ?R = "at_right (0::real)"
```
```  1309   let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
```
```  1310
```
```  1311   show "LIM x ?R. ?G x :> at_top"
```
```  1312     using g_0 unfolding filterlim_at_top_to_right .
```
```  1313
```
```  1314   show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
```
```  1315     unfolding eventually_at_right_to_top
```
```  1316     using Dg eventually_ge_at_top[where c="1::real"]
```
```  1317     apply eventually_elim
```
```  1318     apply (rule DERIV_cong)
```
```  1319     apply (rule DERIV_chain'[where f=inverse])
```
```  1320     apply (auto intro!:  DERIV_inverse)
```
```  1321     done
```
```  1322
```
```  1323   show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
```
```  1324     unfolding eventually_at_right_to_top
```
```  1325     using Df eventually_ge_at_top[where c="1::real"]
```
```  1326     apply eventually_elim
```
```  1327     apply (rule DERIV_cong)
```
```  1328     apply (rule DERIV_chain'[where f=inverse])
```
```  1329     apply (auto intro!:  DERIV_inverse)
```
```  1330     done
```
```  1331
```
```  1332   show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
```
```  1333     unfolding eventually_at_right_to_top
```
```  1334     using g' eventually_ge_at_top[where c="1::real"]
```
```  1335     by eventually_elim auto
```
```  1336
```
```  1337   show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
```
```  1338     unfolding filterlim_at_right_to_top
```
```  1339     apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
```
```  1340     using eventually_ge_at_top[where c="1::real"]
```
```  1341     by eventually_elim simp
```
```  1342 qed
```
```  1343
```
```  1344 end
```