src/HOL/Deriv.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 50347 77e3effa50b6 child 51476 0c0efde246d1 permissions -rw-r--r--
introduce order topology
```     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 Lim
```
```    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 primrec
```
```    23   Bolzano_bisect :: "(real \<times> real \<Rightarrow> bool) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real \<times> real" where
```
```    24   "Bolzano_bisect P a b 0 = (a, b)"
```
```    25   | "Bolzano_bisect P a b (Suc n) =
```
```    26       (let (x, y) = Bolzano_bisect P a b n
```
```    27        in if P (x, (x+y) / 2) then ((x+y)/2, y)
```
```    28                               else (x, (x+y)/2))"
```
```    29
```
```    30
```
```    31 subsection {* Derivatives *}
```
```    32
```
```    33 lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
```
```    34 by (simp add: deriv_def)
```
```    35
```
```    36 lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
```
```    37 by (simp add: deriv_def)
```
```    38
```
```    39 lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
```
```    40   by (simp add: deriv_def tendsto_const)
```
```    41
```
```    42 lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
```
```    43   by (simp add: deriv_def tendsto_const cong: LIM_cong)
```
```    44
```
```    45 lemma DERIV_add:
```
```    46   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
```
```    47   by (simp only: deriv_def add_diff_add add_divide_distrib tendsto_add)
```
```    48
```
```    49 lemma DERIV_minus:
```
```    50   "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
```
```    51   by (simp only: deriv_def minus_diff_minus divide_minus_left tendsto_minus)
```
```    52
```
```    53 lemma DERIV_diff:
```
```    54   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
```
```    55 by (simp only: diff_minus DERIV_add DERIV_minus)
```
```    56
```
```    57 lemma DERIV_add_minus:
```
```    58   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
```
```    59 by (simp only: DERIV_add DERIV_minus)
```
```    60
```
```    61 lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
```
```    62 proof (unfold isCont_iff)
```
```    63   assume "DERIV f x :> D"
```
```    64   hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"
```
```    65     by (rule DERIV_D)
```
```    66   hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
```
```    67     by (intro tendsto_mult tendsto_ident_at)
```
```    68   hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
```
```    69     by simp
```
```    70   hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
```
```    71     by (simp cong: LIM_cong)
```
```    72   thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
```
```    73     by (simp add: LIM_def dist_norm)
```
```    74 qed
```
```    75
```
```    76 lemma DERIV_mult_lemma:
```
```    77   fixes a b c d :: "'a::real_field"
```
```    78   shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"
```
```    79   by (simp add: field_simps diff_divide_distrib)
```
```    80
```
```    81 lemma DERIV_mult':
```
```    82   assumes f: "DERIV f x :> D"
```
```    83   assumes g: "DERIV g x :> E"
```
```    84   shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
```
```    85 proof (unfold deriv_def)
```
```    86   from f have "isCont f x"
```
```    87     by (rule DERIV_isCont)
```
```    88   hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
```
```    89     by (simp only: isCont_iff)
```
```    90   hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +
```
```    91               ((f(x+h) - f x) / h) * g x)
```
```    92           -- 0 --> f x * E + D * g x"
```
```    93     by (intro tendsto_intros DERIV_D f g)
```
```    94   thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)
```
```    95          -- 0 --> f x * E + D * g x"
```
```    96     by (simp only: DERIV_mult_lemma)
```
```    97 qed
```
```    98
```
```    99 lemma DERIV_mult:
```
```   100     "DERIV f x :> Da \<Longrightarrow> DERIV g x :> Db \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x :> Da * g x + Db * f x"
```
```   101   by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
```
```   102
```
```   103 lemma DERIV_unique:
```
```   104     "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
```
```   105   unfolding deriv_def by (rule LIM_unique)
```
```   106
```
```   107 text{*Differentiation of finite sum*}
```
```   108
```
```   109 lemma DERIV_setsum:
```
```   110   assumes "finite S"
```
```   111   and "\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x :> (f' x n)"
```
```   112   shows "DERIV (%x. setsum (f x) S) x :> setsum (f' x) S"
```
```   113   using assms by induct (auto intro!: DERIV_add)
```
```   114
```
```   115 lemma DERIV_sumr [rule_format (no_asm)]:
```
```   116      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
```
```   117       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
```
```   118   by (auto intro: DERIV_setsum)
```
```   119
```
```   120 text{*Alternative definition for differentiability*}
```
```   121
```
```   122 lemma DERIV_LIM_iff:
```
```   123   fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
```
```   124      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
```
```   125       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
```
```   126 apply (rule iffI)
```
```   127 apply (drule_tac k="- a" in LIM_offset)
```
```   128 apply (simp add: diff_minus)
```
```   129 apply (drule_tac k="a" in LIM_offset)
```
```   130 apply (simp add: add_commute)
```
```   131 done
```
```   132
```
```   133 lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
```
```   134 by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
```
```   135
```
```   136 lemma DERIV_inverse_lemma:
```
```   137   "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
```
```   138    \<Longrightarrow> (inverse a - inverse b) / h
```
```   139      = - (inverse a * ((a - b) / h) * inverse b)"
```
```   140 by (simp add: inverse_diff_inverse)
```
```   141
```
```   142 lemma DERIV_inverse':
```
```   143   assumes der: "DERIV f x :> D"
```
```   144   assumes neq: "f x \<noteq> 0"
```
```   145   shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
```
```   146     (is "DERIV _ _ :> ?E")
```
```   147 proof (unfold DERIV_iff2)
```
```   148   from der have lim_f: "f -- x --> f x"
```
```   149     by (rule DERIV_isCont [unfolded isCont_def])
```
```   150
```
```   151   from neq have "0 < norm (f x)" by simp
```
```   152   with LIM_D [OF lim_f] obtain s
```
```   153     where s: "0 < s"
```
```   154     and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
```
```   155                   \<Longrightarrow> norm (f z - f x) < norm (f x)"
```
```   156     by fast
```
```   157
```
```   158   show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
```
```   159   proof (rule LIM_equal2 [OF s])
```
```   160     fix z
```
```   161     assume "z \<noteq> x" "norm (z - x) < s"
```
```   162     hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
```
```   163     hence "f z \<noteq> 0" by auto
```
```   164     thus "(inverse (f z) - inverse (f x)) / (z - x) =
```
```   165           - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"
```
```   166       using neq by (rule DERIV_inverse_lemma)
```
```   167   next
```
```   168     from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"
```
```   169       by (unfold DERIV_iff2)
```
```   170     thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
```
```   171           -- x --> ?E"
```
```   172       by (intro tendsto_intros lim_f neq)
```
```   173   qed
```
```   174 qed
```
```   175
```
```   176 lemma DERIV_divide:
```
```   177   "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
```
```   178    \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
```
```   179 apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
```
```   180           D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
```
```   181 apply (erule subst)
```
```   182 apply (unfold divide_inverse)
```
```   183 apply (erule DERIV_mult')
```
```   184 apply (erule (1) DERIV_inverse')
```
```   185 apply (simp add: ring_distribs nonzero_inverse_mult_distrib)
```
```   186 done
```
```   187
```
```   188 lemma DERIV_power_Suc:
```
```   189   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"
```
```   190   assumes f: "DERIV f x :> D"
```
```   191   shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"
```
```   192 proof (induct n)
```
```   193 case 0
```
```   194   show ?case by (simp add: f)
```
```   195 case (Suc k)
```
```   196   from DERIV_mult' [OF f Suc] show ?case
```
```   197     apply (simp only: of_nat_Suc ring_distribs mult_1_left)
```
```   198     apply (simp only: power_Suc algebra_simps)
```
```   199     done
```
```   200 qed
```
```   201
```
```   202 lemma DERIV_power:
```
```   203   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"
```
```   204   assumes f: "DERIV f x :> D"
```
```   205   shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
```
```   206 by (cases "n", simp, simp add: DERIV_power_Suc f del: power_Suc)
```
```   207
```
```   208 text {* Caratheodory formulation of derivative at a point *}
```
```   209
```
```   210 lemma CARAT_DERIV:
```
```   211      "(DERIV f x :> l) =
```
```   212       (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
```
```   213       (is "?lhs = ?rhs")
```
```   214 proof
```
```   215   assume der: "DERIV f x :> l"
```
```   216   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
```
```   217   proof (intro exI conjI)
```
```   218     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
```
```   219     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
```
```   220     show "isCont ?g x" using der
```
```   221       by (simp add: isCont_iff DERIV_iff diff_minus
```
```   222                cong: LIM_equal [rule_format])
```
```   223     show "?g x = l" by simp
```
```   224   qed
```
```   225 next
```
```   226   assume "?rhs"
```
```   227   then obtain g where
```
```   228     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
```
```   229   thus "(DERIV f x :> l)"
```
```   230      by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
```
```   231 qed
```
```   232
```
```   233 lemma DERIV_chain':
```
```   234   assumes f: "DERIV f x :> D"
```
```   235   assumes g: "DERIV g (f x) :> E"
```
```   236   shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
```
```   237 proof (unfold DERIV_iff2)
```
```   238   obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
```
```   239     and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
```
```   240     using CARAT_DERIV [THEN iffD1, OF g] by fast
```
```   241   from f have "f -- x --> f x"
```
```   242     by (rule DERIV_isCont [unfolded isCont_def])
```
```   243   with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
```
```   244     by (rule isCont_tendsto_compose)
```
```   245   hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
```
```   246           -- x --> d (f x) * D"
```
```   247     by (rule tendsto_mult [OF _ f [unfolded DERIV_iff2]])
```
```   248   thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
```
```   249     by (simp add: d dfx)
```
```   250 qed
```
```   251
```
```   252 text {*
```
```   253  Let's do the standard proof, though theorem
```
```   254  @{text "LIM_mult2"} follows from a NS proof
```
```   255 *}
```
```   256
```
```   257 lemma DERIV_cmult:
```
```   258       "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
```
```   259 by (drule DERIV_mult' [OF DERIV_const], simp)
```
```   260
```
```   261 lemma DERIV_cdivide: "DERIV f x :> D ==> DERIV (%x. f x / c) x :> D / c"
```
```   262   apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x :> (1 / c) * D", force)
```
```   263   apply (erule DERIV_cmult)
```
```   264   done
```
```   265
```
```   266 text {* Standard version *}
```
```   267 lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
```
```   268 by (drule (1) DERIV_chain', simp add: o_def mult_commute)
```
```   269
```
```   270 lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
```
```   271 by (auto dest: DERIV_chain simp add: o_def)
```
```   272
```
```   273 text {* Derivative of linear multiplication *}
```
```   274 lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
```
```   275 by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
```
```   276
```
```   277 lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
```
```   278 apply (cut_tac DERIV_power [OF DERIV_ident])
```
```   279 apply (simp add: real_of_nat_def)
```
```   280 done
```
```   281
```
```   282 text {* Power of @{text "-1"} *}
```
```   283
```
```   284 lemma DERIV_inverse:
```
```   285   fixes x :: "'a::{real_normed_field}"
```
```   286   shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
```
```   287 by (drule DERIV_inverse' [OF DERIV_ident]) simp
```
```   288
```
```   289 text {* Derivative of inverse *}
```
```   290 lemma DERIV_inverse_fun:
```
```   291   fixes x :: "'a::{real_normed_field}"
```
```   292   shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
```
```   293       ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
```
```   294 by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
```
```   295
```
```   296 text {* Derivative of quotient *}
```
```   297 lemma DERIV_quotient:
```
```   298   fixes x :: "'a::{real_normed_field}"
```
```   299   shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
```
```   300        ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
```
```   301 by (drule (2) DERIV_divide) (simp add: mult_commute)
```
```   302
```
```   303 text {* @{text "DERIV_intros"} *}
```
```   304 ML {*
```
```   305 structure Deriv_Intros = Named_Thms
```
```   306 (
```
```   307   val name = @{binding DERIV_intros}
```
```   308   val description = "DERIV introduction rules"
```
```   309 )
```
```   310 *}
```
```   311
```
```   312 setup Deriv_Intros.setup
```
```   313
```
```   314 lemma DERIV_cong: "\<lbrakk> DERIV f x :> X ; X = Y \<rbrakk> \<Longrightarrow> DERIV f x :> Y"
```
```   315   by simp
```
```   316
```
```   317 declare
```
```   318   DERIV_const[THEN DERIV_cong, DERIV_intros]
```
```   319   DERIV_ident[THEN DERIV_cong, DERIV_intros]
```
```   320   DERIV_add[THEN DERIV_cong, DERIV_intros]
```
```   321   DERIV_minus[THEN DERIV_cong, DERIV_intros]
```
```   322   DERIV_mult[THEN DERIV_cong, DERIV_intros]
```
```   323   DERIV_diff[THEN DERIV_cong, DERIV_intros]
```
```   324   DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
```
```   325   DERIV_divide[THEN DERIV_cong, DERIV_intros]
```
```   326   DERIV_power[where 'a=real, THEN DERIV_cong,
```
```   327               unfolded real_of_nat_def[symmetric], DERIV_intros]
```
```   328   DERIV_setsum[THEN DERIV_cong, DERIV_intros]
```
```   329
```
```   330
```
```   331 subsection {* Differentiability predicate *}
```
```   332
```
```   333 definition
```
```   334   differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
```
```   335     (infixl "differentiable" 60) where
```
```   336   "f differentiable x = (\<exists>D. DERIV f x :> D)"
```
```   337
```
```   338 lemma differentiableE [elim?]:
```
```   339   assumes "f differentiable x"
```
```   340   obtains df where "DERIV f x :> df"
```
```   341   using assms unfolding differentiable_def ..
```
```   342
```
```   343 lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
```
```   344 by (simp add: differentiable_def)
```
```   345
```
```   346 lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
```
```   347 by (force simp add: differentiable_def)
```
```   348
```
```   349 lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
```
```   350   by (rule DERIV_ident [THEN differentiableI])
```
```   351
```
```   352 lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
```
```   353   by (rule DERIV_const [THEN differentiableI])
```
```   354
```
```   355 lemma differentiable_compose:
```
```   356   assumes f: "f differentiable (g x)"
```
```   357   assumes g: "g differentiable x"
```
```   358   shows "(\<lambda>x. f (g x)) differentiable x"
```
```   359 proof -
```
```   360   from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
```
```   361   moreover
```
```   362   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   363   ultimately
```
```   364   have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
```
```   365   thus ?thesis by (rule differentiableI)
```
```   366 qed
```
```   367
```
```   368 lemma differentiable_sum [simp]:
```
```   369   assumes "f differentiable x"
```
```   370   and "g differentiable x"
```
```   371   shows "(\<lambda>x. f x + g x) differentiable x"
```
```   372 proof -
```
```   373   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   374   moreover
```
```   375   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   376   ultimately
```
```   377   have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
```
```   378   thus ?thesis by (rule differentiableI)
```
```   379 qed
```
```   380
```
```   381 lemma differentiable_minus [simp]:
```
```   382   assumes "f differentiable x"
```
```   383   shows "(\<lambda>x. - f x) differentiable x"
```
```   384 proof -
```
```   385   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   386   hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
```
```   387   thus ?thesis by (rule differentiableI)
```
```   388 qed
```
```   389
```
```   390 lemma differentiable_diff [simp]:
```
```   391   assumes "f differentiable x"
```
```   392   assumes "g differentiable x"
```
```   393   shows "(\<lambda>x. f x - g x) differentiable x"
```
```   394   unfolding diff_minus using assms by simp
```
```   395
```
```   396 lemma differentiable_mult [simp]:
```
```   397   assumes "f differentiable x"
```
```   398   assumes "g differentiable x"
```
```   399   shows "(\<lambda>x. f x * g x) differentiable x"
```
```   400 proof -
```
```   401   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   402   moreover
```
```   403   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   404   ultimately
```
```   405   have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
```
```   406   thus ?thesis by (rule differentiableI)
```
```   407 qed
```
```   408
```
```   409 lemma differentiable_inverse [simp]:
```
```   410   assumes "f differentiable x" and "f x \<noteq> 0"
```
```   411   shows "(\<lambda>x. inverse (f x)) differentiable x"
```
```   412 proof -
```
```   413   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   414   hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
```
```   415     using `f x \<noteq> 0` by (rule DERIV_inverse')
```
```   416   thus ?thesis by (rule differentiableI)
```
```   417 qed
```
```   418
```
```   419 lemma differentiable_divide [simp]:
```
```   420   assumes "f differentiable x"
```
```   421   assumes "g differentiable x" and "g x \<noteq> 0"
```
```   422   shows "(\<lambda>x. f x / g x) differentiable x"
```
```   423   unfolding divide_inverse using assms by simp
```
```   424
```
```   425 lemma differentiable_power [simp]:
```
```   426   fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a"
```
```   427   assumes "f differentiable x"
```
```   428   shows "(\<lambda>x. f x ^ n) differentiable x"
```
```   429   apply (induct n)
```
```   430   apply simp
```
```   431   apply (simp add: assms)
```
```   432   done
```
```   433
```
```   434
```
```   435 subsection {* Nested Intervals and Bisection *}
```
```   436
```
```   437 text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
```
```   438      All considerably tidied by lcp.*}
```
```   439
```
```   440 lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"
```
```   441 apply (induct "no")
```
```   442 apply (auto intro: order_trans)
```
```   443 done
```
```   444
```
```   445 lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   446          \<forall>n. g(Suc n) \<le> g(n);
```
```   447          \<forall>n. f(n) \<le> g(n) |]
```
```   448       ==> Bseq (f :: nat \<Rightarrow> real)"
```
```   449 apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
```
```   450 apply (rule conjI)
```
```   451 apply (induct_tac "n")
```
```   452 apply (auto intro: order_trans)
```
```   453 apply (rule_tac y = "g n" in order_trans)
```
```   454 apply (induct_tac [2] "n")
```
```   455 apply (auto intro: order_trans)
```
```   456 done
```
```   457
```
```   458 lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   459          \<forall>n. g(Suc n) \<le> g(n);
```
```   460          \<forall>n. f(n) \<le> g(n) |]
```
```   461       ==> Bseq (g :: nat \<Rightarrow> real)"
```
```   462 apply (subst Bseq_minus_iff [symmetric])
```
```   463 apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
```
```   464 apply auto
```
```   465 done
```
```   466
```
```   467 lemma f_inc_imp_le_lim:
```
```   468   fixes f :: "nat \<Rightarrow> real"
```
```   469   shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
```
```   470   by (rule incseq_le, simp add: incseq_SucI, simp add: convergent_LIMSEQ_iff)
```
```   471
```
```   472 lemma lim_uminus:
```
```   473   fixes g :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   474   shows "convergent g ==> lim (%x. - g x) = - (lim g)"
```
```   475 apply (rule tendsto_minus [THEN limI])
```
```   476 apply (simp add: convergent_LIMSEQ_iff)
```
```   477 done
```
```   478
```
```   479 lemma g_dec_imp_lim_le:
```
```   480   fixes g :: "nat \<Rightarrow> real"
```
```   481   shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
```
```   482   by (rule decseq_le, simp add: decseq_SucI, simp add: convergent_LIMSEQ_iff)
```
```   483
```
```   484 lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   485          \<forall>n. g(Suc n) \<le> g(n);
```
```   486          \<forall>n. f(n) \<le> g(n) |]
```
```   487       ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
```
```   488                             ((\<forall>n. m \<le> g(n)) & g ----> m)"
```
```   489 apply (subgoal_tac "monoseq f & monoseq g")
```
```   490 prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
```
```   491 apply (subgoal_tac "Bseq f & Bseq g")
```
```   492 prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
```
```   493 apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
```
```   494 apply (rule_tac x = "lim f" in exI)
```
```   495 apply (rule_tac x = "lim g" in exI)
```
```   496 apply (auto intro: LIMSEQ_le)
```
```   497 apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
```
```   498 done
```
```   499
```
```   500 lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   501          \<forall>n. g(Suc n) \<le> g(n);
```
```   502          \<forall>n. f(n) \<le> g(n);
```
```   503          (%n. f(n) - g(n)) ----> 0 |]
```
```   504       ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
```
```   505                 ((\<forall>n. l \<le> g(n)) & g ----> l)"
```
```   506 apply (drule lemma_nest, auto)
```
```   507 apply (subgoal_tac "l = m")
```
```   508 apply (drule_tac [2] f = f in tendsto_diff)
```
```   509 apply (auto intro: LIMSEQ_unique)
```
```   510 done
```
```   511
```
```   512 text{*The universal quantifiers below are required for the declaration
```
```   513   of @{text Bolzano_nest_unique} below.*}
```
```   514
```
```   515 lemma Bolzano_bisect_le:
```
```   516  "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
```
```   517 apply (rule allI)
```
```   518 apply (induct_tac "n")
```
```   519 apply (auto simp add: Let_def split_def)
```
```   520 done
```
```   521
```
```   522 lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
```
```   523    \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
```
```   524 apply (rule allI)
```
```   525 apply (induct_tac "n")
```
```   526 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
```
```   527 done
```
```   528
```
```   529 lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
```
```   530    \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
```
```   531 apply (rule allI)
```
```   532 apply (induct_tac "n")
```
```   533 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
```
```   534 done
```
```   535
```
```   536 lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
```
```   537 apply (auto)
```
```   538 apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
```
```   539 apply (simp)
```
```   540 done
```
```   541
```
```   542 lemma Bolzano_bisect_diff:
```
```   543      "a \<le> b ==>
```
```   544       snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
```
```   545       (b-a) / (2 ^ n)"
```
```   546 apply (induct "n")
```
```   547 apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
```
```   548 done
```
```   549
```
```   550 lemmas Bolzano_nest_unique =
```
```   551     lemma_nest_unique
```
```   552     [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
```
```   553
```
```   554
```
```   555 lemma not_P_Bolzano_bisect:
```
```   556   assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
```
```   557       and notP: "~ P(a,b)"
```
```   558       and le:   "a \<le> b"
```
```   559   shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
```
```   560 proof (induct n)
```
```   561   case 0 show ?case using notP by simp
```
```   562  next
```
```   563   case (Suc n)
```
```   564   thus ?case
```
```   565  by (auto simp del: surjective_pairing [symmetric]
```
```   566              simp add: Let_def split_def Bolzano_bisect_le [OF le]
```
```   567      P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
```
```   568 qed
```
```   569
```
```   570 (*Now we re-package P_prem as a formula*)
```
```   571 lemma not_P_Bolzano_bisect':
```
```   572      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
```
```   573          ~ P(a,b);  a \<le> b |] ==>
```
```   574       \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
```
```   575 by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
```
```   576
```
```   577
```
```   578
```
```   579 lemma lemma_BOLZANO:
```
```   580      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
```
```   581          \<forall>x. \<exists>d::real. 0 < d &
```
```   582                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
```
```   583          a \<le> b |]
```
```   584       ==> P(a,b)"
```
```   585 apply (rule Bolzano_nest_unique [where P=P, THEN exE], assumption+)
```
```   586 apply (rule tendsto_minus_cancel)
```
```   587 apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
```
```   588 apply (rule ccontr)
```
```   589 apply (drule not_P_Bolzano_bisect', assumption+)
```
```   590 apply (rename_tac "l")
```
```   591 apply (drule_tac x = l in spec, clarify)
```
```   592 apply (simp add: LIMSEQ_iff)
```
```   593 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
```
```   594 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
```
```   595 apply (drule real_less_half_sum, auto)
```
```   596 apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
```
```   597 apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
```
```   598 apply safe
```
```   599 apply (simp_all (no_asm_simp))
```
```   600 apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans)
```
```   601 apply (simp (no_asm_simp) add: abs_if)
```
```   602 apply (rule real_sum_of_halves [THEN subst])
```
```   603 apply (rule add_strict_mono)
```
```   604 apply (simp_all add: diff_minus [symmetric])
```
```   605 done
```
```   606
```
```   607
```
```   608 lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
```
```   609        (\<forall>x. \<exists>d::real. 0 < d &
```
```   610                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
```
```   611       --> (\<forall>a b. a \<le> b --> P(a,b))"
```
```   612 apply clarify
```
```   613 apply (blast intro: lemma_BOLZANO)
```
```   614 done
```
```   615
```
```   616
```
```   617 subsection {* Intermediate Value Theorem *}
```
```   618
```
```   619 text {*Prove Contrapositive by Bisection*}
```
```   620
```
```   621 lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
```
```   622          a \<le> b;
```
```   623          (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
```
```   624       ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
```
```   625 apply (rule contrapos_pp, assumption)
```
```   626 apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
```
```   627 apply safe
```
```   628 apply simp_all
```
```   629 apply (simp add: isCont_iff LIM_eq)
```
```   630 apply (rule ccontr)
```
```   631 apply (subgoal_tac "a \<le> x & x \<le> b")
```
```   632  prefer 2
```
```   633  apply simp
```
```   634  apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
```
```   635 apply (drule_tac x = x in spec)+
```
```   636 apply simp
```
```   637 apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
```
```   638 apply safe
```
```   639 apply simp
```
```   640 apply (drule_tac x = s in spec, clarify)
```
```   641 apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
```
```   642 apply (drule_tac x = "ba-x" in spec)
```
```   643 apply (simp_all add: abs_if)
```
```   644 apply (drule_tac x = "aa-x" in spec)
```
```   645 apply (case_tac "x \<le> aa", simp_all)
```
```   646 done
```
```   647
```
```   648 lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
```
```   649          a \<le> b;
```
```   650          (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
```
```   651       |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
```
```   652 apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
```
```   653 apply (drule IVT [where f = "%x. - f x"], assumption)
```
```   654 apply simp_all
```
```   655 done
```
```   656
```
```   657 (*HOL style here: object-level formulations*)
```
```   658 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
```
```   659       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```   660       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```   661 apply (blast intro: IVT)
```
```   662 done
```
```   663
```
```   664 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
```
```   665       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```   666       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```   667 apply (blast intro: IVT2)
```
```   668 done
```
```   669
```
```   670
```
```   671 subsection {* Boundedness of continuous functions *}
```
```   672
```
```   673 text{*By bisection, function continuous on closed interval is bounded above*}
```
```   674
```
```   675 lemma isCont_bounded:
```
```   676      "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   677       ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
```
```   678 apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M)" in lemma_BOLZANO2)
```
```   679 apply safe
```
```   680 apply simp_all
```
```   681 apply (rename_tac x xa ya M Ma)
```
```   682 apply (metis linorder_not_less order_le_less order_trans)
```
```   683 apply (case_tac "a \<le> x & x \<le> b")
```
```   684  prefer 2
```
```   685  apply (rule_tac x = 1 in exI, force)
```
```   686 apply (simp add: LIM_eq isCont_iff)
```
```   687 apply (drule_tac x = x in spec, auto)
```
```   688 apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
```
```   689 apply (drule_tac x = 1 in spec, auto)
```
```   690 apply (rule_tac x = s in exI, clarify)
```
```   691 apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
```
```   692 apply (drule_tac x = "xa-x" in spec)
```
```   693 apply (auto simp add: abs_ge_self)
```
```   694 done
```
```   695
```
```   696 text{*Refine the above to existence of least upper bound*}
```
```   697
```
```   698 lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
```
```   699       (\<exists>t. isLub UNIV S t)"
```
```   700 by (blast intro: reals_complete)
```
```   701
```
```   702 lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   703          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
```
```   704                    (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
```
```   705 apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
```
```   706         in lemma_reals_complete)
```
```   707 apply auto
```
```   708 apply (drule isCont_bounded, assumption)
```
```   709 apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
```
```   710 apply (rule exI, auto)
```
```   711 apply (auto dest!: spec simp add: linorder_not_less)
```
```   712 done
```
```   713
```
```   714 text{*Now show that it attains its upper bound*}
```
```   715
```
```   716 lemma isCont_eq_Ub:
```
```   717   assumes le: "a \<le> b"
```
```   718       and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
```
```   719   shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
```
```   720              (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
```
```   721 proof -
```
```   722   from isCont_has_Ub [OF le con]
```
```   723   obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
```
```   724              and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
```
```   725   show ?thesis
```
```   726   proof (intro exI, intro conjI)
```
```   727     show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
```
```   728     show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
```
```   729     proof (rule ccontr)
```
```   730       assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
```
```   731       with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
```
```   732         by (fastforce simp add: linorder_not_le [symmetric])
```
```   733       hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
```
```   734         by (auto simp add: con)
```
```   735       from isCont_bounded [OF le this]
```
```   736       obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
```
```   737       have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
```
```   738         by (simp add: M3 algebra_simps)
```
```   739       have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
```
```   740         by (auto intro: order_le_less_trans [of _ k])
```
```   741       with Minv
```
```   742       have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
```
```   743         by (intro strip less_imp_inverse_less, simp_all)
```
```   744       hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
```
```   745         by simp
```
```   746       have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
```
```   747         by (simp, arith)
```
```   748       from M2 [OF this]
```
```   749       obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
```
```   750       thus False using invlt [of x] by force
```
```   751     qed
```
```   752   qed
```
```   753 qed
```
```   754
```
```   755
```
```   756 text{*Same theorem for lower bound*}
```
```   757
```
```   758 lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   759          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
```
```   760                    (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
```
```   761 apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
```
```   762 prefer 2 apply (blast intro: isCont_minus)
```
```   763 apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
```
```   764 apply safe
```
```   765 apply auto
```
```   766 done
```
```   767
```
```   768
```
```   769 text{*Another version.*}
```
```   770
```
```   771 lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   772       ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
```
```   773           (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
```
```   774 apply (frule isCont_eq_Lb)
```
```   775 apply (frule_tac [2] isCont_eq_Ub)
```
```   776 apply (assumption+, safe)
```
```   777 apply (rule_tac x = "f x" in exI)
```
```   778 apply (rule_tac x = "f xa" in exI, simp, safe)
```
```   779 apply (cut_tac x = x and y = xa in linorder_linear, safe)
```
```   780 apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
```
```   781 apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
```
```   782 apply (rule_tac [2] x = xb in exI)
```
```   783 apply (rule_tac [4] x = xb in exI, simp_all)
```
```   784 done
```
```   785
```
```   786
```
```   787 subsection {* Local extrema *}
```
```   788
```
```   789 text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
```
```   790
```
```   791 lemma DERIV_pos_inc_right:
```
```   792   fixes f :: "real => real"
```
```   793   assumes der: "DERIV f x :> l"
```
```   794       and l:   "0 < l"
```
```   795   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
```
```   796 proof -
```
```   797   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
```
```   798   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)"
```
```   799     by (simp add: diff_minus)
```
```   800   then obtain s
```
```   801         where s:   "0 < s"
```
```   802           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
```
```   803     by auto
```
```   804   thus ?thesis
```
```   805   proof (intro exI conjI strip)
```
```   806     show "0<s" using s .
```
```   807     fix h::real
```
```   808     assume "0 < h" "h < s"
```
```   809     with all [of h] show "f x < f (x+h)"
```
```   810     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   811     split add: split_if_asm)
```
```   812       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
```
```   813       with l
```
```   814       have "0 < (f (x+h) - f x) / h" by arith
```
```   815       thus "f x < f (x+h)"
```
```   816   by (simp add: pos_less_divide_eq h)
```
```   817     qed
```
```   818   qed
```
```   819 qed
```
```   820
```
```   821 lemma DERIV_neg_dec_left:
```
```   822   fixes f :: "real => real"
```
```   823   assumes der: "DERIV f x :> l"
```
```   824       and l:   "l < 0"
```
```   825   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
```
```   826 proof -
```
```   827   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
```
```   828   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)"
```
```   829     by (simp add: diff_minus)
```
```   830   then obtain s
```
```   831         where s:   "0 < s"
```
```   832           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
```
```   833     by auto
```
```   834   thus ?thesis
```
```   835   proof (intro exI conjI strip)
```
```   836     show "0<s" using s .
```
```   837     fix h::real
```
```   838     assume "0 < h" "h < s"
```
```   839     with all [of "-h"] show "f x < f (x-h)"
```
```   840     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   841     split add: split_if_asm)
```
```   842       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
```
```   843       with l
```
```   844       have "0 < (f (x-h) - f x) / h" by arith
```
```   845       thus "f x < f (x-h)"
```
```   846   by (simp add: pos_less_divide_eq h)
```
```   847     qed
```
```   848   qed
```
```   849 qed
```
```   850
```
```   851
```
```   852 lemma DERIV_pos_inc_left:
```
```   853   fixes f :: "real => real"
```
```   854   shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
```
```   855   apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])
```
```   856   apply (auto simp add: DERIV_minus)
```
```   857   done
```
```   858
```
```   859 lemma DERIV_neg_dec_right:
```
```   860   fixes f :: "real => real"
```
```   861   shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
```
```   862   apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])
```
```   863   apply (auto simp add: DERIV_minus)
```
```   864   done
```
```   865
```
```   866 lemma DERIV_local_max:
```
```   867   fixes f :: "real => real"
```
```   868   assumes der: "DERIV f x :> l"
```
```   869       and d:   "0 < d"
```
```   870       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
```
```   871   shows "l = 0"
```
```   872 proof (cases rule: linorder_cases [of l 0])
```
```   873   case equal thus ?thesis .
```
```   874 next
```
```   875   case less
```
```   876   from DERIV_neg_dec_left [OF der less]
```
```   877   obtain d' where d': "0 < d'"
```
```   878              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
```
```   879   from real_lbound_gt_zero [OF d d']
```
```   880   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   881   with lt le [THEN spec [where x="x-e"]]
```
```   882   show ?thesis by (auto simp add: abs_if)
```
```   883 next
```
```   884   case greater
```
```   885   from DERIV_pos_inc_right [OF der greater]
```
```   886   obtain d' where d': "0 < d'"
```
```   887              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
```
```   888   from real_lbound_gt_zero [OF d d']
```
```   889   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   890   with lt le [THEN spec [where x="x+e"]]
```
```   891   show ?thesis by (auto simp add: abs_if)
```
```   892 qed
```
```   893
```
```   894
```
```   895 text{*Similar theorem for a local minimum*}
```
```   896 lemma DERIV_local_min:
```
```   897   fixes f :: "real => real"
```
```   898   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
```
```   899 by (drule DERIV_minus [THEN DERIV_local_max], auto)
```
```   900
```
```   901
```
```   902 text{*In particular, if a function is locally flat*}
```
```   903 lemma DERIV_local_const:
```
```   904   fixes f :: "real => real"
```
```   905   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
```
```   906 by (auto dest!: DERIV_local_max)
```
```   907
```
```   908
```
```   909 subsection {* Rolle's Theorem *}
```
```   910
```
```   911 text{*Lemma about introducing open ball in open interval*}
```
```   912 lemma lemma_interval_lt:
```
```   913      "[| a < x;  x < b |]
```
```   914       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
```
```   915
```
```   916 apply (simp add: abs_less_iff)
```
```   917 apply (insert linorder_linear [of "x-a" "b-x"], safe)
```
```   918 apply (rule_tac x = "x-a" in exI)
```
```   919 apply (rule_tac [2] x = "b-x" in exI, auto)
```
```   920 done
```
```   921
```
```   922 lemma lemma_interval: "[| a < x;  x < b |] ==>
```
```   923         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
```
```   924 apply (drule lemma_interval_lt, auto)
```
```   925 apply force
```
```   926 done
```
```   927
```
```   928 text{*Rolle's Theorem.
```
```   929    If @{term f} is defined and continuous on the closed interval
```
```   930    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
```
```   931    and @{term "f(a) = f(b)"},
```
```   932    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
```
```   933 theorem Rolle:
```
```   934   assumes lt: "a < b"
```
```   935       and eq: "f(a) = f(b)"
```
```   936       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```   937       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```   938   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
```
```   939 proof -
```
```   940   have le: "a \<le> b" using lt by simp
```
```   941   from isCont_eq_Ub [OF le con]
```
```   942   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
```
```   943              and alex: "a \<le> x" and xleb: "x \<le> b"
```
```   944     by blast
```
```   945   from isCont_eq_Lb [OF le con]
```
```   946   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
```
```   947               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
```
```   948     by blast
```
```   949   show ?thesis
```
```   950   proof cases
```
```   951     assume axb: "a < x & x < b"
```
```   952         --{*@{term f} attains its maximum within the interval*}
```
```   953     hence ax: "a<x" and xb: "x<b" by arith +
```
```   954     from lemma_interval [OF ax xb]
```
```   955     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   956       by blast
```
```   957     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
```
```   958       by blast
```
```   959     from differentiableD [OF dif [OF axb]]
```
```   960     obtain l where der: "DERIV f x :> l" ..
```
```   961     have "l=0" by (rule DERIV_local_max [OF der d bound'])
```
```   962         --{*the derivative at a local maximum is zero*}
```
```   963     thus ?thesis using ax xb der by auto
```
```   964   next
```
```   965     assume notaxb: "~ (a < x & x < b)"
```
```   966     hence xeqab: "x=a | x=b" using alex xleb by arith
```
```   967     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
```
```   968     show ?thesis
```
```   969     proof cases
```
```   970       assume ax'b: "a < x' & x' < b"
```
```   971         --{*@{term f} attains its minimum within the interval*}
```
```   972       hence ax': "a<x'" and x'b: "x'<b" by arith+
```
```   973       from lemma_interval [OF ax' x'b]
```
```   974       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   975   by blast
```
```   976       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
```
```   977   by blast
```
```   978       from differentiableD [OF dif [OF ax'b]]
```
```   979       obtain l where der: "DERIV f x' :> l" ..
```
```   980       have "l=0" by (rule DERIV_local_min [OF der d bound'])
```
```   981         --{*the derivative at a local minimum is zero*}
```
```   982       thus ?thesis using ax' x'b der by auto
```
```   983     next
```
```   984       assume notax'b: "~ (a < x' & x' < b)"
```
```   985         --{*@{term f} is constant througout the interval*}
```
```   986       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
```
```   987       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
```
```   988       from dense [OF lt]
```
```   989       obtain r where ar: "a < r" and rb: "r < b" by blast
```
```   990       from lemma_interval [OF ar rb]
```
```   991       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   992   by blast
```
```   993       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
```
```   994       proof (clarify)
```
```   995         fix z::real
```
```   996         assume az: "a \<le> z" and zb: "z \<le> b"
```
```   997         show "f z = f b"
```
```   998         proof (rule order_antisym)
```
```   999           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
```
```  1000           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
```
```  1001         qed
```
```  1002       qed
```
```  1003       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
```
```  1004       proof (intro strip)
```
```  1005         fix y::real
```
```  1006         assume lt: "\<bar>r-y\<bar> < d"
```
```  1007         hence "f y = f b" by (simp add: eq_fb bound)
```
```  1008         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
```
```  1009       qed
```
```  1010       from differentiableD [OF dif [OF conjI [OF ar rb]]]
```
```  1011       obtain l where der: "DERIV f r :> l" ..
```
```  1012       have "l=0" by (rule DERIV_local_const [OF der d bound'])
```
```  1013         --{*the derivative of a constant function is zero*}
```
```  1014       thus ?thesis using ar rb der by auto
```
```  1015     qed
```
```  1016   qed
```
```  1017 qed
```
```  1018
```
```  1019
```
```  1020 subsection{*Mean Value Theorem*}
```
```  1021
```
```  1022 lemma lemma_MVT:
```
```  1023      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
```
```  1024 proof cases
```
```  1025   assume "a=b" thus ?thesis by simp
```
```  1026 next
```
```  1027   assume "a\<noteq>b"
```
```  1028   hence ba: "b-a \<noteq> 0" by arith
```
```  1029   show ?thesis
```
```  1030     by (rule real_mult_left_cancel [OF ba, THEN iffD1],
```
```  1031         simp add: right_diff_distrib,
```
```  1032         simp add: left_diff_distrib)
```
```  1033 qed
```
```  1034
```
```  1035 theorem MVT:
```
```  1036   assumes lt:  "a < b"
```
```  1037       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```  1038       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```  1039   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
```
```  1040                    (f(b) - f(a) = (b-a) * l)"
```
```  1041 proof -
```
```  1042   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
```
```  1043   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
```
```  1044     using con by (fast intro: isCont_intros)
```
```  1045   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
```
```  1046   proof (clarify)
```
```  1047     fix x::real
```
```  1048     assume ax: "a < x" and xb: "x < b"
```
```  1049     from differentiableD [OF dif [OF conjI [OF ax xb]]]
```
```  1050     obtain l where der: "DERIV f x :> l" ..
```
```  1051     show "?F differentiable x"
```
```  1052       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
```
```  1053           blast intro: DERIV_diff DERIV_cmult_Id der)
```
```  1054   qed
```
```  1055   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
```
```  1056   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
```
```  1057     by blast
```
```  1058   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
```
```  1059     by (rule DERIV_cmult_Id)
```
```  1060   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
```
```  1061                    :> 0 + (f b - f a) / (b - a)"
```
```  1062     by (rule DERIV_add [OF der])
```
```  1063   show ?thesis
```
```  1064   proof (intro exI conjI)
```
```  1065     show "a < z" using az .
```
```  1066     show "z < b" using zb .
```
```  1067     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
```
```  1068     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
```
```  1069   qed
```
```  1070 qed
```
```  1071
```
```  1072 lemma MVT2:
```
```  1073      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
```
```  1074       ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
```
```  1075 apply (drule MVT)
```
```  1076 apply (blast intro: DERIV_isCont)
```
```  1077 apply (force dest: order_less_imp_le simp add: differentiable_def)
```
```  1078 apply (blast dest: DERIV_unique order_less_imp_le)
```
```  1079 done
```
```  1080
```
```  1081
```
```  1082 text{*A function is constant if its derivative is 0 over an interval.*}
```
```  1083
```
```  1084 lemma DERIV_isconst_end:
```
```  1085   fixes f :: "real => real"
```
```  1086   shows "[| a < b;
```
```  1087          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1088          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1089         ==> f b = f a"
```
```  1090 apply (drule MVT, assumption)
```
```  1091 apply (blast intro: differentiableI)
```
```  1092 apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
```
```  1093 done
```
```  1094
```
```  1095 lemma DERIV_isconst1:
```
```  1096   fixes f :: "real => real"
```
```  1097   shows "[| a < b;
```
```  1098          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1099          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1100         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
```
```  1101 apply safe
```
```  1102 apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
```
```  1103 apply (drule_tac b = x in DERIV_isconst_end, auto)
```
```  1104 done
```
```  1105
```
```  1106 lemma DERIV_isconst2:
```
```  1107   fixes f :: "real => real"
```
```  1108   shows "[| a < b;
```
```  1109          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1110          \<forall>x. a < x & x < b --> DERIV f x :> 0;
```
```  1111          a \<le> x; x \<le> b |]
```
```  1112         ==> f x = f a"
```
```  1113 apply (blast dest: DERIV_isconst1)
```
```  1114 done
```
```  1115
```
```  1116 lemma DERIV_isconst3: fixes a b x y :: real
```
```  1117   assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
```
```  1118   assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
```
```  1119   shows "f x = f y"
```
```  1120 proof (cases "x = y")
```
```  1121   case False
```
```  1122   let ?a = "min x y"
```
```  1123   let ?b = "max x y"
```
```  1124
```
```  1125   have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
```
```  1126   proof (rule allI, rule impI)
```
```  1127     fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
```
```  1128     hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
```
```  1129     hence "z \<in> {a<..<b}" by auto
```
```  1130     thus "DERIV f z :> 0" by (rule derivable)
```
```  1131   qed
```
```  1132   hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
```
```  1133     and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
```
```  1134
```
```  1135   have "?a < ?b" using `x \<noteq> y` by auto
```
```  1136   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
```
```  1137   show ?thesis by auto
```
```  1138 qed auto
```
```  1139
```
```  1140 lemma DERIV_isconst_all:
```
```  1141   fixes f :: "real => real"
```
```  1142   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
```
```  1143 apply (rule linorder_cases [of x y])
```
```  1144 apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
```
```  1145 done
```
```  1146
```
```  1147 lemma DERIV_const_ratio_const:
```
```  1148   fixes f :: "real => real"
```
```  1149   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
```
```  1150 apply (rule linorder_cases [of a b], auto)
```
```  1151 apply (drule_tac [!] f = f in MVT)
```
```  1152 apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
```
```  1153 apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
```
```  1154 done
```
```  1155
```
```  1156 lemma DERIV_const_ratio_const2:
```
```  1157   fixes f :: "real => real"
```
```  1158   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
```
```  1159 apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
```
```  1160 apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
```
```  1161 done
```
```  1162
```
```  1163 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
```
```  1164 by (simp)
```
```  1165
```
```  1166 lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
```
```  1167 by (simp)
```
```  1168
```
```  1169 text{*Gallileo's "trick": average velocity = av. of end velocities*}
```
```  1170
```
```  1171 lemma DERIV_const_average:
```
```  1172   fixes v :: "real => real"
```
```  1173   assumes neq: "a \<noteq> (b::real)"
```
```  1174       and der: "\<forall>x. DERIV v x :> k"
```
```  1175   shows "v ((a + b)/2) = (v a + v b)/2"
```
```  1176 proof (cases rule: linorder_cases [of a b])
```
```  1177   case equal with neq show ?thesis by simp
```
```  1178 next
```
```  1179   case less
```
```  1180   have "(v b - v a) / (b - a) = k"
```
```  1181     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1182   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1183   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
```
```  1184     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1185   ultimately show ?thesis using neq by force
```
```  1186 next
```
```  1187   case greater
```
```  1188   have "(v b - v a) / (b - a) = k"
```
```  1189     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1190   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1191   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
```
```  1192     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1193   ultimately show ?thesis using neq by (force simp add: add_commute)
```
```  1194 qed
```
```  1195
```
```  1196 (* A function with positive derivative is increasing.
```
```  1197    A simple proof using the MVT, by Jeremy Avigad. And variants.
```
```  1198 *)
```
```  1199 lemma DERIV_pos_imp_increasing:
```
```  1200   fixes a::real and b::real and f::"real => real"
```
```  1201   assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
```
```  1202   shows "f a < f b"
```
```  1203 proof (rule ccontr)
```
```  1204   assume f: "~ f a < f b"
```
```  1205   have "EX l z. a < z & z < b & DERIV f z :> l
```
```  1206       & f b - f a = (b - a) * l"
```
```  1207     apply (rule MVT)
```
```  1208       using assms
```
```  1209       apply auto
```
```  1210       apply (metis DERIV_isCont)
```
```  1211      apply (metis differentiableI less_le)
```
```  1212     done
```
```  1213   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
```
```  1214       and "f b - f a = (b - a) * l"
```
```  1215     by auto
```
```  1216   with assms f have "~(l > 0)"
```
```  1217     by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
```
```  1218   with assms z show False
```
```  1219     by (metis DERIV_unique less_le)
```
```  1220 qed
```
```  1221
```
```  1222 lemma DERIV_nonneg_imp_nondecreasing:
```
```  1223   fixes a::real and b::real and f::"real => real"
```
```  1224   assumes "a \<le> b" and
```
```  1225     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
```
```  1226   shows "f a \<le> f b"
```
```  1227 proof (rule ccontr, cases "a = b")
```
```  1228   assume "~ f a \<le> f b" and "a = b"
```
```  1229   then show False by auto
```
```  1230 next
```
```  1231   assume A: "~ f a \<le> f b"
```
```  1232   assume B: "a ~= b"
```
```  1233   with assms have "EX l z. a < z & z < b & DERIV f z :> l
```
```  1234       & f b - f a = (b - a) * l"
```
```  1235     apply -
```
```  1236     apply (rule MVT)
```
```  1237       apply auto
```
```  1238       apply (metis DERIV_isCont)
```
```  1239      apply (metis differentiableI less_le)
```
```  1240     done
```
```  1241   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
```
```  1242       and C: "f b - f a = (b - a) * l"
```
```  1243     by auto
```
```  1244   with A have "a < b" "f b < f a" by auto
```
```  1245   with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
```
```  1246     (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
```
```  1247   with assms z show False
```
```  1248     by (metis DERIV_unique order_less_imp_le)
```
```  1249 qed
```
```  1250
```
```  1251 lemma DERIV_neg_imp_decreasing:
```
```  1252   fixes a::real and b::real and f::"real => real"
```
```  1253   assumes "a < b" and
```
```  1254     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
```
```  1255   shows "f a > f b"
```
```  1256 proof -
```
```  1257   have "(%x. -f x) a < (%x. -f x) b"
```
```  1258     apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
```
```  1259     using assms
```
```  1260     apply auto
```
```  1261     apply (metis DERIV_minus neg_0_less_iff_less)
```
```  1262     done
```
```  1263   thus ?thesis
```
```  1264     by simp
```
```  1265 qed
```
```  1266
```
```  1267 lemma DERIV_nonpos_imp_nonincreasing:
```
```  1268   fixes a::real and b::real and f::"real => real"
```
```  1269   assumes "a \<le> b" and
```
```  1270     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
```
```  1271   shows "f a \<ge> f b"
```
```  1272 proof -
```
```  1273   have "(%x. -f x) a \<le> (%x. -f x) b"
```
```  1274     apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
```
```  1275     using assms
```
```  1276     apply auto
```
```  1277     apply (metis DERIV_minus neg_0_le_iff_le)
```
```  1278     done
```
```  1279   thus ?thesis
```
```  1280     by simp
```
```  1281 qed
```
```  1282
```
```  1283 subsection {* Continuous injective functions *}
```
```  1284
```
```  1285 text{*Dull lemma: an continuous injection on an interval must have a
```
```  1286 strict maximum at an end point, not in the middle.*}
```
```  1287
```
```  1288 lemma lemma_isCont_inj:
```
```  1289   fixes f :: "real \<Rightarrow> real"
```
```  1290   assumes d: "0 < d"
```
```  1291       and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1292       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1293   shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
```
```  1294 proof (rule ccontr)
```
```  1295   assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
```
```  1296   hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
```
```  1297   show False
```
```  1298   proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
```
```  1299     case le
```
```  1300     from d cont all [of "x+d"]
```
```  1301     have flef: "f(x+d) \<le> f x"
```
```  1302      and xlex: "x - d \<le> x"
```
```  1303      and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
```
```  1304        by (auto simp add: abs_if)
```
```  1305     from IVT [OF le flef xlex cont']
```
```  1306     obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
```
```  1307     moreover
```
```  1308     hence "g(f x') = g (f(x+d))" by simp
```
```  1309     ultimately show False using d inj [of x'] inj [of "x+d"]
```
```  1310       by (simp add: abs_le_iff)
```
```  1311   next
```
```  1312     case ge
```
```  1313     from d cont all [of "x-d"]
```
```  1314     have flef: "f(x-d) \<le> f x"
```
```  1315      and xlex: "x \<le> x+d"
```
```  1316      and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
```
```  1317        by (auto simp add: abs_if)
```
```  1318     from IVT2 [OF ge flef xlex cont']
```
```  1319     obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
```
```  1320     moreover
```
```  1321     hence "g(f x') = g (f(x-d))" by simp
```
```  1322     ultimately show False using d inj [of x'] inj [of "x-d"]
```
```  1323       by (simp add: abs_le_iff)
```
```  1324   qed
```
```  1325 qed
```
```  1326
```
```  1327
```
```  1328 text{*Similar version for lower bound.*}
```
```  1329
```
```  1330 lemma lemma_isCont_inj2:
```
```  1331   fixes f g :: "real \<Rightarrow> real"
```
```  1332   shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
```
```  1333         \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
```
```  1334       ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
```
```  1335 apply (insert lemma_isCont_inj
```
```  1336           [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
```
```  1337 apply (simp add: linorder_not_le)
```
```  1338 done
```
```  1339
```
```  1340 text{*Show there's an interval surrounding @{term "f(x)"} in
```
```  1341 @{text "f[[x - d, x + d]]"} .*}
```
```  1342
```
```  1343 lemma isCont_inj_range:
```
```  1344   fixes f :: "real \<Rightarrow> real"
```
```  1345   assumes d: "0 < d"
```
```  1346       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1347       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1348   shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
```
```  1349 proof -
```
```  1350   have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
```
```  1351     by (auto simp add: abs_le_iff)
```
```  1352   from isCont_Lb_Ub [OF this]
```
```  1353   obtain L M
```
```  1354   where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
```
```  1355     and all2 [rule_format]:
```
```  1356            "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
```
```  1357     by auto
```
```  1358   with d have "L \<le> f x & f x \<le> M" by simp
```
```  1359   moreover have "L \<noteq> f x"
```
```  1360   proof -
```
```  1361     from lemma_isCont_inj2 [OF d inj cont]
```
```  1362     obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
```
```  1363     thus ?thesis using all1 [of u] by arith
```
```  1364   qed
```
```  1365   moreover have "f x \<noteq> M"
```
```  1366   proof -
```
```  1367     from lemma_isCont_inj [OF d inj cont]
```
```  1368     obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
```
```  1369     thus ?thesis using all1 [of u] by arith
```
```  1370   qed
```
```  1371   ultimately have "L < f x & f x < M" by arith
```
```  1372   hence "0 < f x - L" "0 < M - f x" by arith+
```
```  1373   from real_lbound_gt_zero [OF this]
```
```  1374   obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
```
```  1375   thus ?thesis
```
```  1376   proof (intro exI conjI)
```
```  1377     show "0<e" using e(1) .
```
```  1378     show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
```
```  1379     proof (intro strip)
```
```  1380       fix y::real
```
```  1381       assume "\<bar>y - f x\<bar> \<le> e"
```
```  1382       with e have "L \<le> y \<and> y \<le> M" by arith
```
```  1383       from all2 [OF this]
```
```  1384       obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
```
```  1385       thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
```
```  1386         by (force simp add: abs_le_iff)
```
```  1387     qed
```
```  1388   qed
```
```  1389 qed
```
```  1390
```
```  1391
```
```  1392 text{*Continuity of inverse function*}
```
```  1393
```
```  1394 lemma isCont_inverse_function:
```
```  1395   fixes f g :: "real \<Rightarrow> real"
```
```  1396   assumes d: "0 < d"
```
```  1397       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1398       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1399   shows "isCont g (f x)"
```
```  1400 proof (simp add: isCont_iff LIM_eq)
```
```  1401   show "\<forall>r. 0 < r \<longrightarrow>
```
```  1402          (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
```
```  1403   proof (intro strip)
```
```  1404     fix r::real
```
```  1405     assume r: "0<r"
```
```  1406     from real_lbound_gt_zero [OF r d]
```
```  1407     obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
```
```  1408     with inj cont
```
```  1409     have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
```
```  1410                   "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
```
```  1411     from isCont_inj_range [OF e this]
```
```  1412     obtain e' where e': "0 < e'"
```
```  1413         and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
```
```  1414           by blast
```
```  1415     show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"
```
```  1416     proof (intro exI conjI)
```
```  1417       show "0<e'" using e' .
```
```  1418       show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
```
```  1419       proof (intro strip)
```
```  1420         fix z::real
```
```  1421         assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
```
```  1422         with e e_lt e_simps all [rule_format, of "f x + z"]
```
```  1423         show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
```
```  1424       qed
```
```  1425     qed
```
```  1426   qed
```
```  1427 qed
```
```  1428
```
```  1429 text {* Derivative of inverse function *}
```
```  1430
```
```  1431 lemma DERIV_inverse_function:
```
```  1432   fixes f g :: "real \<Rightarrow> real"
```
```  1433   assumes der: "DERIV f (g x) :> D"
```
```  1434   assumes neq: "D \<noteq> 0"
```
```  1435   assumes a: "a < x" and b: "x < b"
```
```  1436   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
```
```  1437   assumes cont: "isCont g x"
```
```  1438   shows "DERIV g x :> inverse D"
```
```  1439 unfolding DERIV_iff2
```
```  1440 proof (rule LIM_equal2)
```
```  1441   show "0 < min (x - a) (b - x)"
```
```  1442     using a b by arith
```
```  1443 next
```
```  1444   fix y
```
```  1445   assume "norm (y - x) < min (x - a) (b - x)"
```
```  1446   hence "a < y" and "y < b"
```
```  1447     by (simp_all add: abs_less_iff)
```
```  1448   thus "(g y - g x) / (y - x) =
```
```  1449         inverse ((f (g y) - x) / (g y - g x))"
```
```  1450     by (simp add: inj)
```
```  1451 next
```
```  1452   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
```
```  1453     by (rule der [unfolded DERIV_iff2])
```
```  1454   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
```
```  1455     using inj a b by simp
```
```  1456   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
```
```  1457   proof (safe intro!: exI)
```
```  1458     show "0 < min (x - a) (b - x)"
```
```  1459       using a b by simp
```
```  1460   next
```
```  1461     fix y
```
```  1462     assume "norm (y - x) < min (x - a) (b - x)"
```
```  1463     hence y: "a < y" "y < b"
```
```  1464       by (simp_all add: abs_less_iff)
```
```  1465     assume "g y = g x"
```
```  1466     hence "f (g y) = f (g x)" by simp
```
```  1467     hence "y = x" using inj y a b by simp
```
```  1468     also assume "y \<noteq> x"
```
```  1469     finally show False by simp
```
```  1470   qed
```
```  1471   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
```
```  1472     using cont 1 2 by (rule isCont_LIM_compose2)
```
```  1473   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
```
```  1474         -- x --> inverse D"
```
```  1475     using neq by (rule tendsto_inverse)
```
```  1476 qed
```
```  1477
```
```  1478
```
```  1479 subsection {* Generalized Mean Value Theorem *}
```
```  1480
```
```  1481 theorem GMVT:
```
```  1482   fixes a b :: real
```
```  1483   assumes alb: "a < b"
```
```  1484     and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  1485     and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
```
```  1486     and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
```
```  1487     and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
```
```  1488   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)"
```
```  1489 proof -
```
```  1490   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
```
```  1491   from assms have "a < b" by simp
```
```  1492   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
```
```  1493     using fc gc by simp
```
```  1494   moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
```
```  1495     using fd gd by simp
```
```  1496   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)
```
```  1497   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" ..
```
```  1498   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
```
```  1499
```
```  1500   from cdef have cint: "a < c \<and> c < b" by auto
```
```  1501   with gd have "g differentiable c" by simp
```
```  1502   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
```
```  1503   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
```
```  1504
```
```  1505   from cdef have "a < c \<and> c < b" by auto
```
```  1506   with fd have "f differentiable c" by simp
```
```  1507   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
```
```  1508   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
```
```  1509
```
```  1510   from cdef have "DERIV ?h c :> l" by auto
```
```  1511   moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
```
```  1512     using g'cdef f'cdef by (auto intro!: DERIV_intros)
```
```  1513   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
```
```  1514
```
```  1515   {
```
```  1516     from cdef have "?h b - ?h a = (b - a) * l" by auto
```
```  1517     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1518     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1519   }
```
```  1520   moreover
```
```  1521   {
```
```  1522     have "?h b - ?h a =
```
```  1523          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
```
```  1524           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
```
```  1525       by (simp add: algebra_simps)
```
```  1526     hence "?h b - ?h a = 0" by auto
```
```  1527   }
```
```  1528   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
```
```  1529   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
```
```  1530   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
```
```  1531   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
```
```  1532
```
```  1533   with g'cdef f'cdef cint show ?thesis by auto
```
```  1534 qed
```
```  1535
```
```  1536
```
```  1537 subsection {* Theorems about Limits *}
```
```  1538
```
```  1539 (* need to rename second isCont_inverse *)
```
```  1540
```
```  1541 lemma isCont_inv_fun:
```
```  1542   fixes f g :: "real \<Rightarrow> real"
```
```  1543   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  1544          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  1545       ==> isCont g (f x)"
```
```  1546 by (rule isCont_inverse_function)
```
```  1547
```
```  1548 lemma isCont_inv_fun_inv:
```
```  1549   fixes f g :: "real \<Rightarrow> real"
```
```  1550   shows "[| 0 < d;
```
```  1551          \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  1552          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  1553        ==> \<exists>e. 0 < e &
```
```  1554              (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
```
```  1555 apply (drule isCont_inj_range)
```
```  1556 prefer 2 apply (assumption, assumption, auto)
```
```  1557 apply (rule_tac x = e in exI, auto)
```
```  1558 apply (rotate_tac 2)
```
```  1559 apply (drule_tac x = y in spec, auto)
```
```  1560 done
```
```  1561
```
```  1562
```
```  1563 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
```
```  1564 lemma LIM_fun_gt_zero:
```
```  1565      "[| f -- c --> (l::real); 0 < l |]
```
```  1566          ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
```
```  1567 apply (drule (1) LIM_D, clarify)
```
```  1568 apply (rule_tac x = s in exI)
```
```  1569 apply (simp add: abs_less_iff)
```
```  1570 done
```
```  1571
```
```  1572 lemma LIM_fun_less_zero:
```
```  1573      "[| f -- c --> (l::real); l < 0 |]
```
```  1574       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
```
```  1575 apply (drule LIM_D [where r="-l"], simp, clarify)
```
```  1576 apply (rule_tac x = s in exI)
```
```  1577 apply (simp add: abs_less_iff)
```
```  1578 done
```
```  1579
```
```  1580 lemma LIM_fun_not_zero:
```
```  1581      "[| f -- c --> (l::real); l \<noteq> 0 |]
```
```  1582       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
```
```  1583 apply (rule linorder_cases [of l 0])
```
```  1584 apply (drule (1) LIM_fun_less_zero, force)
```
```  1585 apply simp
```
```  1586 apply (drule (1) LIM_fun_gt_zero, force)
```
```  1587 done
```
```  1588
```
```  1589 lemma GMVT':
```
```  1590   fixes f g :: "real \<Rightarrow> real"
```
```  1591   assumes "a < b"
```
```  1592   assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
```
```  1593   assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
```
```  1594   assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
```
```  1595   assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
```
```  1596   shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
```
```  1597 proof -
```
```  1598   have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
```
```  1599     a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
```
```  1600     using assms by (intro GMVT) (force simp: differentiable_def)+
```
```  1601   then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
```
```  1602     using DERIV_f DERIV_g by (force dest: DERIV_unique)
```
```  1603   then show ?thesis
```
```  1604     by auto
```
```  1605 qed
```
```  1606
```
```  1607 lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
```
```  1608     DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
```
```  1609   unfolding DERIV_iff2
```
```  1610 proof (rule filterlim_cong)
```
```  1611   assume "eventually (\<lambda>x. f x = g x) (nhds x)"
```
```  1612   moreover then have "f x = g x" by (auto simp: eventually_nhds)
```
```  1613   moreover assume "x = y" "u = v"
```
```  1614   ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
```
```  1615     by (auto simp: eventually_within at_def elim: eventually_elim1)
```
```  1616 qed simp_all
```
```  1617
```
```  1618 lemma DERIV_shift:
```
```  1619   "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
```
```  1620   by (simp add: DERIV_iff field_simps)
```
```  1621
```
```  1622 lemma DERIV_mirror:
```
```  1623   "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
```
```  1624   by (simp add: deriv_def filterlim_at_split filterlim_at_left_to_right
```
```  1625                 tendsto_minus_cancel_left field_simps conj_commute)
```
```  1626
```
```  1627 lemma lhopital_right_0:
```
```  1628   fixes f0 g0 :: "real \<Rightarrow> real"
```
```  1629   assumes f_0: "(f0 ---> 0) (at_right 0)"
```
```  1630   assumes g_0: "(g0 ---> 0) (at_right 0)"
```
```  1631   assumes ev:
```
```  1632     "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
```
```  1633     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
```
```  1634     "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
```
```  1635     "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
```
```  1636   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
```
```  1637   shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
```
```  1638 proof -
```
```  1639   def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
```
```  1640   then have "f 0 = 0" by simp
```
```  1641
```
```  1642   def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
```
```  1643   then have "g 0 = 0" by simp
```
```  1644
```
```  1645   have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
```
```  1646       DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
```
```  1647     using ev by eventually_elim auto
```
```  1648   then obtain a where [arith]: "0 < a"
```
```  1649     and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
```
```  1650     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
```
```  1651     and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
```
```  1652     and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
```
```  1653     unfolding eventually_within eventually_at by (auto simp: dist_real_def)
```
```  1654
```
```  1655   have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
```
```  1656     using g0_neq_0 by (simp add: g_def)
```
```  1657
```
```  1658   { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
```
```  1659       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
```
```  1660          (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
```
```  1661   note f = this
```
```  1662
```
```  1663   { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
```
```  1664       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
```
```  1665          (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
```
```  1666   note g = this
```
```  1667
```
```  1668   have "isCont f 0"
```
```  1669     using tendsto_const[of "0::real" "at 0"] f_0
```
```  1670     unfolding isCont_def f_def
```
```  1671     by (intro filterlim_split_at_real)
```
```  1672        (auto elim: eventually_elim1
```
```  1673              simp add: filterlim_def le_filter_def eventually_within eventually_filtermap)
```
```  1674
```
```  1675   have "isCont g 0"
```
```  1676     using tendsto_const[of "0::real" "at 0"] g_0
```
```  1677     unfolding isCont_def g_def
```
```  1678     by (intro filterlim_split_at_real)
```
```  1679        (auto elim: eventually_elim1
```
```  1680              simp add: filterlim_def le_filter_def eventually_within eventually_filtermap)
```
```  1681
```
```  1682   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)"
```
```  1683   proof (rule bchoice, rule)
```
```  1684     fix x assume "x \<in> {0 <..< a}"
```
```  1685     then have x[arith]: "0 < x" "x < a" by auto
```
```  1686     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"
```
```  1687       by auto
```
```  1688     have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
```
```  1689       using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
```
```  1690     moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
```
```  1691       using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
```
```  1692     ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
```
```  1693       using f g `x < a` by (intro GMVT') auto
```
```  1694     then guess c ..
```
```  1695     moreover
```
```  1696     with g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
```
```  1697       by (simp add: field_simps)
```
```  1698     ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
```
```  1699       using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
```
```  1700   qed
```
```  1701   then guess \<zeta> ..
```
```  1702   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)"
```
```  1703     unfolding eventually_within eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
```
```  1704   moreover
```
```  1705   from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
```
```  1706     by eventually_elim auto
```
```  1707   then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
```
```  1708     by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
```
```  1709        (auto intro: tendsto_const tendsto_ident_at_within)
```
```  1710   then have "(\<zeta> ---> 0) (at_right 0)"
```
```  1711     by (rule tendsto_norm_zero_cancel)
```
```  1712   with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
```
```  1713     by (auto elim!: eventually_elim1 simp: filterlim_within filterlim_at)
```
```  1714   from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
```
```  1715     by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
```
```  1716   ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
```
```  1717     by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
```
```  1718        (auto elim: eventually_elim1)
```
```  1719   also have "?P \<longleftrightarrow> ?thesis"
```
```  1720     by (rule filterlim_cong) (auto simp: f_def g_def eventually_within)
```
```  1721   finally show ?thesis .
```
```  1722 qed
```
```  1723
```
```  1724 lemma lhopital_right:
```
```  1725   "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
```
```  1726     eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1727     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1728     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
```
```  1729     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
```
```  1730     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
```
```  1731   ((\<lambda> x. f x / g x) ---> y) (at_right x)"
```
```  1732   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
```
```  1733   by (rule lhopital_right_0)
```
```  1734
```
```  1735 lemma lhopital_left:
```
```  1736   "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
```
```  1737     eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1738     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1739     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
```
```  1740     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
```
```  1741     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
```
```  1742   ((\<lambda> x. f x / g x) ---> y) (at_left x)"
```
```  1743   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
```
```  1744   by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
```
```  1745
```
```  1746 lemma lhopital:
```
```  1747   "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
```
```  1748     eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1749     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1750     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
```
```  1751     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
```
```  1752     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
```
```  1753   ((\<lambda> x. f x / g x) ---> y) (at x)"
```
```  1754   unfolding eventually_at_split filterlim_at_split
```
```  1755   by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
```
```  1756
```
```  1757 lemma lhopital_right_0_at_top:
```
```  1758   fixes f g :: "real \<Rightarrow> real"
```
```  1759   assumes g_0: "LIM x at_right 0. g x :> at_top"
```
```  1760   assumes ev:
```
```  1761     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
```
```  1762     "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
```
```  1763     "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
```
```  1764   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
```
```  1765   shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
```
```  1766   unfolding tendsto_iff
```
```  1767 proof safe
```
```  1768   fix e :: real assume "0 < e"
```
```  1769
```
```  1770   with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
```
```  1771   have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
```
```  1772   from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
```
```  1773   obtain a where [arith]: "0 < a"
```
```  1774     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
```
```  1775     and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
```
```  1776     and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
```
```  1777     and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
```
```  1778     unfolding eventually_within_le by (auto simp: dist_real_def)
```
```  1779
```
```  1780   from Df have
```
```  1781     "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
```
```  1782     unfolding eventually_within eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
```
```  1783
```
```  1784   moreover
```
```  1785   have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
```
```  1786     using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
```
```  1787
```
```  1788   moreover
```
```  1789   have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
```
```  1790     using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
```
```  1791     by (rule filterlim_compose)
```
```  1792   then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
```
```  1793     by (intro tendsto_intros)
```
```  1794   then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
```
```  1795     by (simp add: inverse_eq_divide)
```
```  1796   from this[unfolded tendsto_iff, rule_format, of 1]
```
```  1797   have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
```
```  1798     by (auto elim!: eventually_elim1 simp: dist_real_def)
```
```  1799
```
```  1800   moreover
```
```  1801   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)"
```
```  1802     by (intro tendsto_intros)
```
```  1803   then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
```
```  1804     by (simp add: inverse_eq_divide)
```
```  1805   from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
```
```  1806   have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
```
```  1807     by (auto simp: dist_real_def)
```
```  1808
```
```  1809   ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
```
```  1810   proof eventually_elim
```
```  1811     fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
```
```  1812     assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
```
```  1813
```
```  1814     have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
```
```  1815       using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
```
```  1816     then guess y ..
```
```  1817     from this
```
```  1818     have [arith]: "t < y" "y < a" and D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
```
```  1819       using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
```
```  1820
```
```  1821     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"
```
```  1822       by (simp add: field_simps)
```
```  1823     have "norm (f t / g t - x) \<le>
```
```  1824         norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
```
```  1825       unfolding * by (rule norm_triangle_ineq)
```
```  1826     also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
```
```  1827       by (simp add: abs_mult D_eq dist_real_def)
```
```  1828     also have "\<dots> < (e / 4) * 2 + e / 2"
```
```  1829       using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
```
```  1830     finally show "dist (f t / g t) x < e"
```
```  1831       by (simp add: dist_real_def)
```
```  1832   qed
```
```  1833 qed
```
```  1834
```
```  1835 lemma lhopital_right_at_top:
```
```  1836   "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1837     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
```
```  1838     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
```
```  1839     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
```
```  1840     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
```
```  1841     ((\<lambda> x. f x / g x) ---> y) (at_right x)"
```
```  1842   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
```
```  1843   by (rule lhopital_right_0_at_top)
```
```  1844
```
```  1845 lemma lhopital_left_at_top:
```
```  1846   "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1847     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
```
```  1848     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
```
```  1849     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
```
```  1850     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
```
```  1851     ((\<lambda> x. f x / g x) ---> y) (at_left x)"
```
```  1852   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
```
```  1853   by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
```
```  1854
```
```  1855 lemma lhopital_at_top:
```
```  1856   "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
```
```  1857     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
```
```  1858     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
```
```  1859     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
```
```  1860     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
```
```  1861     ((\<lambda> x. f x / g x) ---> y) (at x)"
```
```  1862   unfolding eventually_at_split filterlim_at_split
```
```  1863   by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
```
```  1864
```
```  1865 lemma lhospital_at_top_at_top:
```
```  1866   fixes f g :: "real \<Rightarrow> real"
```
```  1867   assumes g_0: "LIM x at_top. g x :> at_top"
```
```  1868   assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
```
```  1869   assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
```
```  1870   assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
```
```  1871   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
```
```  1872   shows "((\<lambda> x. f x / g x) ---> x) at_top"
```
```  1873   unfolding filterlim_at_top_to_right
```
```  1874 proof (rule lhopital_right_0_at_top)
```
```  1875   let ?F = "\<lambda>x. f (inverse x)"
```
```  1876   let ?G = "\<lambda>x. g (inverse x)"
```
```  1877   let ?R = "at_right (0::real)"
```
```  1878   let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
```
```  1879
```
```  1880   show "LIM x ?R. ?G x :> at_top"
```
```  1881     using g_0 unfolding filterlim_at_top_to_right .
```
```  1882
```
```  1883   show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
```
```  1884     unfolding eventually_at_right_to_top
```
```  1885     using Dg eventually_ge_at_top[where c="1::real"]
```
```  1886     apply eventually_elim
```
```  1887     apply (rule DERIV_cong)
```
```  1888     apply (rule DERIV_chain'[where f=inverse])
```
```  1889     apply (auto intro!:  DERIV_inverse)
```
```  1890     done
```
```  1891
```
```  1892   show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
```
```  1893     unfolding eventually_at_right_to_top
```
```  1894     using Df eventually_ge_at_top[where c="1::real"]
```
```  1895     apply eventually_elim
```
```  1896     apply (rule DERIV_cong)
```
```  1897     apply (rule DERIV_chain'[where f=inverse])
```
```  1898     apply (auto intro!:  DERIV_inverse)
```
```  1899     done
```
```  1900
```
```  1901   show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
```
```  1902     unfolding eventually_at_right_to_top
```
```  1903     using g' eventually_ge_at_top[where c="1::real"]
```
```  1904     by eventually_elim auto
```
```  1905
```
```  1906   show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
```
```  1907     unfolding filterlim_at_right_to_top
```
```  1908     apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
```
```  1909     using eventually_ge_at_top[where c="1::real"]
```
```  1910     by eventually_elim simp
```
```  1911 qed
```
```  1912
```
```  1913 end
```