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