src/HOL/Deriv.thy
 author paulson Fri Nov 13 11:33:33 2009 +0000 (2009-11-13) changeset 33654 abf780db30ea parent 31902 862ae16a799d child 33659 2d7ab9458518 permissions -rw-r--r--
A number of theorems contributed by Jeremy Avigad
```     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 consts
```
```    23   Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
```
```    24 primrec
```
```    25   "Bolzano_bisect P a b 0 = (a,b)"
```
```    26   "Bolzano_bisect P a b (Suc n) =
```
```    27       (let (x,y) = Bolzano_bisect P a b n
```
```    28        in if P(x, (x+y)/2) then ((x+y)/2, y)
```
```    29                             else (x, (x+y)/2))"
```
```    30
```
```    31
```
```    32 subsection {* Derivatives *}
```
```    33
```
```    34 lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
```
```    35 by (simp add: deriv_def)
```
```    36
```
```    37 lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
```
```    38 by (simp add: deriv_def)
```
```    39
```
```    40 lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
```
```    41 by (simp add: deriv_def)
```
```    42
```
```    43 lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
```
```    44 by (simp add: deriv_def cong: LIM_cong)
```
```    45
```
```    46 lemma add_diff_add:
```
```    47   fixes a b c d :: "'a::ab_group_add"
```
```    48   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```    49 by simp
```
```    50
```
```    51 lemma DERIV_add:
```
```    52   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
```
```    53 by (simp only: deriv_def add_diff_add add_divide_distrib LIM_add)
```
```    54
```
```    55 lemma DERIV_minus:
```
```    56   "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
```
```    57 by (simp only: deriv_def minus_diff_minus divide_minus_left LIM_minus)
```
```    58
```
```    59 lemma DERIV_diff:
```
```    60   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
```
```    61 by (simp only: diff_def DERIV_add DERIV_minus)
```
```    62
```
```    63 lemma DERIV_add_minus:
```
```    64   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
```
```    65 by (simp only: DERIV_add DERIV_minus)
```
```    66
```
```    67 lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
```
```    68 proof (unfold isCont_iff)
```
```    69   assume "DERIV f x :> D"
```
```    70   hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"
```
```    71     by (rule DERIV_D)
```
```    72   hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
```
```    73     by (intro LIM_mult LIM_ident)
```
```    74   hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
```
```    75     by simp
```
```    76   hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
```
```    77     by (simp cong: LIM_cong)
```
```    78   thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
```
```    79     by (simp add: LIM_def dist_norm)
```
```    80 qed
```
```    81
```
```    82 lemma DERIV_mult_lemma:
```
```    83   fixes a b c d :: "'a::real_field"
```
```    84   shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"
```
```    85 by (simp add: diff_minus add_divide_distrib [symmetric] ring_distribs)
```
```    86
```
```    87 lemma DERIV_mult':
```
```    88   assumes f: "DERIV f x :> D"
```
```    89   assumes g: "DERIV g x :> E"
```
```    90   shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
```
```    91 proof (unfold deriv_def)
```
```    92   from f have "isCont f x"
```
```    93     by (rule DERIV_isCont)
```
```    94   hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
```
```    95     by (simp only: isCont_iff)
```
```    96   hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +
```
```    97               ((f(x+h) - f x) / h) * g x)
```
```    98           -- 0 --> f x * E + D * g x"
```
```    99     by (intro LIM_add LIM_mult LIM_const DERIV_D f g)
```
```   100   thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)
```
```   101          -- 0 --> f x * E + D * g x"
```
```   102     by (simp only: DERIV_mult_lemma)
```
```   103 qed
```
```   104
```
```   105 lemma DERIV_mult:
```
```   106      "[| DERIV f x :> Da; DERIV g x :> Db |]
```
```   107       ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
```
```   108 by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
```
```   109
```
```   110 lemma DERIV_unique:
```
```   111       "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
```
```   112 apply (simp add: deriv_def)
```
```   113 apply (blast intro: LIM_unique)
```
```   114 done
```
```   115
```
```   116 text{*Differentiation of finite sum*}
```
```   117
```
```   118 lemma DERIV_setsum:
```
```   119   assumes "finite S"
```
```   120   and "\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x :> (f' x n)"
```
```   121   shows "DERIV (%x. setsum (f x) S) x :> setsum (f' x) S"
```
```   122   using assms by induct (auto intro!: DERIV_add)
```
```   123
```
```   124 lemma DERIV_sumr [rule_format (no_asm)]:
```
```   125      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
```
```   126       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
```
```   127   by (auto intro: DERIV_setsum)
```
```   128
```
```   129 text{*Alternative definition for differentiability*}
```
```   130
```
```   131 lemma DERIV_LIM_iff:
```
```   132   fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
```
```   133      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
```
```   134       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
```
```   135 apply (rule iffI)
```
```   136 apply (drule_tac k="- a" in LIM_offset)
```
```   137 apply (simp add: diff_minus)
```
```   138 apply (drule_tac k="a" in LIM_offset)
```
```   139 apply (simp add: add_commute)
```
```   140 done
```
```   141
```
```   142 lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
```
```   143 by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
```
```   144
```
```   145 lemma inverse_diff_inverse:
```
```   146   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   147    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   148 by (simp add: algebra_simps)
```
```   149
```
```   150 lemma DERIV_inverse_lemma:
```
```   151   "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
```
```   152    \<Longrightarrow> (inverse a - inverse b) / h
```
```   153      = - (inverse a * ((a - b) / h) * inverse b)"
```
```   154 by (simp add: inverse_diff_inverse)
```
```   155
```
```   156 lemma DERIV_inverse':
```
```   157   assumes der: "DERIV f x :> D"
```
```   158   assumes neq: "f x \<noteq> 0"
```
```   159   shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
```
```   160     (is "DERIV _ _ :> ?E")
```
```   161 proof (unfold DERIV_iff2)
```
```   162   from der have lim_f: "f -- x --> f x"
```
```   163     by (rule DERIV_isCont [unfolded isCont_def])
```
```   164
```
```   165   from neq have "0 < norm (f x)" by simp
```
```   166   with LIM_D [OF lim_f] obtain s
```
```   167     where s: "0 < s"
```
```   168     and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
```
```   169                   \<Longrightarrow> norm (f z - f x) < norm (f x)"
```
```   170     by fast
```
```   171
```
```   172   show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
```
```   173   proof (rule LIM_equal2 [OF s])
```
```   174     fix z
```
```   175     assume "z \<noteq> x" "norm (z - x) < s"
```
```   176     hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
```
```   177     hence "f z \<noteq> 0" by auto
```
```   178     thus "(inverse (f z) - inverse (f x)) / (z - x) =
```
```   179           - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"
```
```   180       using neq by (rule DERIV_inverse_lemma)
```
```   181   next
```
```   182     from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"
```
```   183       by (unfold DERIV_iff2)
```
```   184     thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
```
```   185           -- x --> ?E"
```
```   186       by (intro LIM_mult LIM_inverse LIM_minus LIM_const lim_f neq)
```
```   187   qed
```
```   188 qed
```
```   189
```
```   190 lemma DERIV_divide:
```
```   191   "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
```
```   192    \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
```
```   193 apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
```
```   194           D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
```
```   195 apply (erule subst)
```
```   196 apply (unfold divide_inverse)
```
```   197 apply (erule DERIV_mult')
```
```   198 apply (erule (1) DERIV_inverse')
```
```   199 apply (simp add: ring_distribs nonzero_inverse_mult_distrib)
```
```   200 apply (simp add: mult_ac)
```
```   201 done
```
```   202
```
```   203 lemma DERIV_power_Suc:
```
```   204   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"
```
```   205   assumes f: "DERIV f x :> D"
```
```   206   shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"
```
```   207 proof (induct n)
```
```   208 case 0
```
```   209   show ?case by (simp add: f)
```
```   210 case (Suc k)
```
```   211   from DERIV_mult' [OF f Suc] show ?case
```
```   212     apply (simp only: of_nat_Suc ring_distribs mult_1_left)
```
```   213     apply (simp only: power_Suc algebra_simps)
```
```   214     done
```
```   215 qed
```
```   216
```
```   217 lemma DERIV_power:
```
```   218   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"
```
```   219   assumes f: "DERIV f x :> D"
```
```   220   shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
```
```   221 by (cases "n", simp, simp add: DERIV_power_Suc f del: power_Suc)
```
```   222
```
```   223 text {* Caratheodory formulation of derivative at a point *}
```
```   224
```
```   225 lemma CARAT_DERIV:
```
```   226      "(DERIV f x :> l) =
```
```   227       (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
```
```   228       (is "?lhs = ?rhs")
```
```   229 proof
```
```   230   assume der: "DERIV f x :> l"
```
```   231   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
```
```   232   proof (intro exI conjI)
```
```   233     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
```
```   234     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
```
```   235     show "isCont ?g x" using der
```
```   236       by (simp add: isCont_iff DERIV_iff diff_minus
```
```   237                cong: LIM_equal [rule_format])
```
```   238     show "?g x = l" by simp
```
```   239   qed
```
```   240 next
```
```   241   assume "?rhs"
```
```   242   then obtain g where
```
```   243     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
```
```   244   thus "(DERIV f x :> l)"
```
```   245      by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
```
```   246 qed
```
```   247
```
```   248 lemma DERIV_chain':
```
```   249   assumes f: "DERIV f x :> D"
```
```   250   assumes g: "DERIV g (f x) :> E"
```
```   251   shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
```
```   252 proof (unfold DERIV_iff2)
```
```   253   obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
```
```   254     and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
```
```   255     using CARAT_DERIV [THEN iffD1, OF g] by fast
```
```   256   from f have "f -- x --> f x"
```
```   257     by (rule DERIV_isCont [unfolded isCont_def])
```
```   258   with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
```
```   259     by (rule isCont_LIM_compose)
```
```   260   hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
```
```   261           -- x --> d (f x) * D"
```
```   262     by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
```
```   263   thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
```
```   264     by (simp add: d dfx real_scaleR_def)
```
```   265 qed
```
```   266
```
```   267 text {*
```
```   268  Let's do the standard proof, though theorem
```
```   269  @{text "LIM_mult2"} follows from a NS proof
```
```   270 *}
```
```   271
```
```   272 lemma DERIV_cmult:
```
```   273       "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
```
```   274 by (drule DERIV_mult' [OF DERIV_const], simp)
```
```   275
```
```   276 lemma DERIV_cdivide: "DERIV f x :> D ==> DERIV (%x. f x / c) x :> D / c"
```
```   277   apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x :> (1 / c) * D", force)
```
```   278   apply (erule DERIV_cmult)
```
```   279   done
```
```   280
```
```   281 text {* Standard version *}
```
```   282 lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
```
```   283 by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
```
```   284
```
```   285 lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
```
```   286 by (auto dest: DERIV_chain simp add: o_def)
```
```   287
```
```   288 text {* Derivative of linear multiplication *}
```
```   289 lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
```
```   290 by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
```
```   291
```
```   292 lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
```
```   293 apply (cut_tac DERIV_power [OF DERIV_ident])
```
```   294 apply (simp add: real_scaleR_def real_of_nat_def)
```
```   295 done
```
```   296
```
```   297 text {* Power of @{text "-1"} *}
```
```   298
```
```   299 lemma DERIV_inverse:
```
```   300   fixes x :: "'a::{real_normed_field}"
```
```   301   shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
```
```   302 by (drule DERIV_inverse' [OF DERIV_ident]) simp
```
```   303
```
```   304 text {* Derivative of inverse *}
```
```   305 lemma DERIV_inverse_fun:
```
```   306   fixes x :: "'a::{real_normed_field}"
```
```   307   shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
```
```   308       ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
```
```   309 by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
```
```   310
```
```   311 text {* Derivative of quotient *}
```
```   312 lemma DERIV_quotient:
```
```   313   fixes x :: "'a::{real_normed_field}"
```
```   314   shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
```
```   315        ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
```
```   316 by (drule (2) DERIV_divide) (simp add: mult_commute)
```
```   317
```
```   318 lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
```
```   319 by auto
```
```   320
```
```   321 text {* @{text "DERIV_intros"} *}
```
```   322 ML {*
```
```   323 structure Deriv_Intros = Named_Thms
```
```   324 (
```
```   325   val name = "DERIV_intros"
```
```   326   val description = "DERIV introduction rules"
```
```   327 )
```
```   328 *}
```
```   329
```
```   330 setup Deriv_Intros.setup
```
```   331
```
```   332 lemma DERIV_cong: "\<lbrakk> DERIV f x :> X ; X = Y \<rbrakk> \<Longrightarrow> DERIV f x :> Y"
```
```   333   by simp
```
```   334
```
```   335 declare
```
```   336   DERIV_const[THEN DERIV_cong, DERIV_intros]
```
```   337   DERIV_ident[THEN DERIV_cong, DERIV_intros]
```
```   338   DERIV_add[THEN DERIV_cong, DERIV_intros]
```
```   339   DERIV_minus[THEN DERIV_cong, DERIV_intros]
```
```   340   DERIV_mult[THEN DERIV_cong, DERIV_intros]
```
```   341   DERIV_diff[THEN DERIV_cong, DERIV_intros]
```
```   342   DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
```
```   343   DERIV_divide[THEN DERIV_cong, DERIV_intros]
```
```   344   DERIV_power[where 'a=real, THEN DERIV_cong,
```
```   345               unfolded real_of_nat_def[symmetric], DERIV_intros]
```
```   346   DERIV_setsum[THEN DERIV_cong, DERIV_intros]
```
```   347
```
```   348
```
```   349 subsection {* Differentiability predicate *}
```
```   350
```
```   351 definition
```
```   352   differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
```
```   353     (infixl "differentiable" 60) where
```
```   354   "f differentiable x = (\<exists>D. DERIV f x :> D)"
```
```   355
```
```   356 lemma differentiableE [elim?]:
```
```   357   assumes "f differentiable x"
```
```   358   obtains df where "DERIV f x :> df"
```
```   359   using prems unfolding differentiable_def ..
```
```   360
```
```   361 lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
```
```   362 by (simp add: differentiable_def)
```
```   363
```
```   364 lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
```
```   365 by (force simp add: differentiable_def)
```
```   366
```
```   367 lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
```
```   368   by (rule DERIV_ident [THEN differentiableI])
```
```   369
```
```   370 lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
```
```   371   by (rule DERIV_const [THEN differentiableI])
```
```   372
```
```   373 lemma differentiable_compose:
```
```   374   assumes f: "f differentiable (g x)"
```
```   375   assumes g: "g differentiable x"
```
```   376   shows "(\<lambda>x. f (g x)) differentiable x"
```
```   377 proof -
```
```   378   from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
```
```   379   moreover
```
```   380   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   381   ultimately
```
```   382   have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
```
```   383   thus ?thesis by (rule differentiableI)
```
```   384 qed
```
```   385
```
```   386 lemma differentiable_sum [simp]:
```
```   387   assumes "f differentiable x"
```
```   388   and "g differentiable x"
```
```   389   shows "(\<lambda>x. f x + g x) differentiable x"
```
```   390 proof -
```
```   391   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   392   moreover
```
```   393   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   394   ultimately
```
```   395   have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
```
```   396   thus ?thesis by (rule differentiableI)
```
```   397 qed
```
```   398
```
```   399 lemma differentiable_minus [simp]:
```
```   400   assumes "f differentiable x"
```
```   401   shows "(\<lambda>x. - f x) differentiable x"
```
```   402 proof -
```
```   403   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   404   hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
```
```   405   thus ?thesis by (rule differentiableI)
```
```   406 qed
```
```   407
```
```   408 lemma differentiable_diff [simp]:
```
```   409   assumes "f differentiable x"
```
```   410   assumes "g differentiable x"
```
```   411   shows "(\<lambda>x. f x - g x) differentiable x"
```
```   412   unfolding diff_minus using prems by simp
```
```   413
```
```   414 lemma differentiable_mult [simp]:
```
```   415   assumes "f differentiable x"
```
```   416   assumes "g differentiable x"
```
```   417   shows "(\<lambda>x. f x * g x) differentiable x"
```
```   418 proof -
```
```   419   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   420   moreover
```
```   421   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   422   ultimately
```
```   423   have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
```
```   424   thus ?thesis by (rule differentiableI)
```
```   425 qed
```
```   426
```
```   427 lemma differentiable_inverse [simp]:
```
```   428   assumes "f differentiable x" and "f x \<noteq> 0"
```
```   429   shows "(\<lambda>x. inverse (f x)) differentiable x"
```
```   430 proof -
```
```   431   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   432   hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
```
```   433     using `f x \<noteq> 0` by (rule DERIV_inverse')
```
```   434   thus ?thesis by (rule differentiableI)
```
```   435 qed
```
```   436
```
```   437 lemma differentiable_divide [simp]:
```
```   438   assumes "f differentiable x"
```
```   439   assumes "g differentiable x" and "g x \<noteq> 0"
```
```   440   shows "(\<lambda>x. f x / g x) differentiable x"
```
```   441   unfolding divide_inverse using prems by simp
```
```   442
```
```   443 lemma differentiable_power [simp]:
```
```   444   fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a"
```
```   445   assumes "f differentiable x"
```
```   446   shows "(\<lambda>x. f x ^ n) differentiable x"
```
```   447   by (induct n, simp, simp add: prems)
```
```   448
```
```   449
```
```   450 subsection {* Nested Intervals and Bisection *}
```
```   451
```
```   452 text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
```
```   453      All considerably tidied by lcp.*}
```
```   454
```
```   455 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)"
```
```   456 apply (induct "no")
```
```   457 apply (auto intro: order_trans)
```
```   458 done
```
```   459
```
```   460 lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   461          \<forall>n. g(Suc n) \<le> g(n);
```
```   462          \<forall>n. f(n) \<le> g(n) |]
```
```   463       ==> Bseq (f :: nat \<Rightarrow> real)"
```
```   464 apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
```
```   465 apply (induct_tac "n")
```
```   466 apply (auto intro: order_trans)
```
```   467 apply (rule_tac y = "g (Suc na)" in order_trans)
```
```   468 apply (induct_tac [2] "na")
```
```   469 apply (auto intro: order_trans)
```
```   470 done
```
```   471
```
```   472 lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   473          \<forall>n. g(Suc n) \<le> g(n);
```
```   474          \<forall>n. f(n) \<le> g(n) |]
```
```   475       ==> Bseq (g :: nat \<Rightarrow> real)"
```
```   476 apply (subst Bseq_minus_iff [symmetric])
```
```   477 apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
```
```   478 apply auto
```
```   479 done
```
```   480
```
```   481 lemma f_inc_imp_le_lim:
```
```   482   fixes f :: "nat \<Rightarrow> real"
```
```   483   shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
```
```   484 apply (rule linorder_not_less [THEN iffD1])
```
```   485 apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
```
```   486 apply (drule real_less_sum_gt_zero)
```
```   487 apply (drule_tac x = "f n + - lim f" in spec, safe)
```
```   488 apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
```
```   489 apply (subgoal_tac "lim f \<le> f (no + n) ")
```
```   490 apply (drule_tac no=no and m=n in lemma_f_mono_add)
```
```   491 apply (auto simp add: add_commute)
```
```   492 apply (induct_tac "no")
```
```   493 apply simp
```
```   494 apply (auto intro: order_trans simp add: diff_minus abs_if)
```
```   495 done
```
```   496
```
```   497 lemma lim_uminus:
```
```   498   fixes g :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```   499   shows "convergent g ==> lim (%x. - g x) = - (lim g)"
```
```   500 apply (rule LIMSEQ_minus [THEN limI])
```
```   501 apply (simp add: convergent_LIMSEQ_iff)
```
```   502 done
```
```   503
```
```   504 lemma g_dec_imp_lim_le:
```
```   505   fixes g :: "nat \<Rightarrow> real"
```
```   506   shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
```
```   507 apply (subgoal_tac "- (g n) \<le> - (lim g) ")
```
```   508 apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
```
```   509 apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
```
```   510 done
```
```   511
```
```   512 lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   513          \<forall>n. g(Suc n) \<le> g(n);
```
```   514          \<forall>n. f(n) \<le> g(n) |]
```
```   515       ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
```
```   516                             ((\<forall>n. m \<le> g(n)) & g ----> m)"
```
```   517 apply (subgoal_tac "monoseq f & monoseq g")
```
```   518 prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
```
```   519 apply (subgoal_tac "Bseq f & Bseq g")
```
```   520 prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
```
```   521 apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
```
```   522 apply (rule_tac x = "lim f" in exI)
```
```   523 apply (rule_tac x = "lim g" in exI)
```
```   524 apply (auto intro: LIMSEQ_le)
```
```   525 apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
```
```   526 done
```
```   527
```
```   528 lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   529          \<forall>n. g(Suc n) \<le> g(n);
```
```   530          \<forall>n. f(n) \<le> g(n);
```
```   531          (%n. f(n) - g(n)) ----> 0 |]
```
```   532       ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
```
```   533                 ((\<forall>n. l \<le> g(n)) & g ----> l)"
```
```   534 apply (drule lemma_nest, auto)
```
```   535 apply (subgoal_tac "l = m")
```
```   536 apply (drule_tac [2] X = f in LIMSEQ_diff)
```
```   537 apply (auto intro: LIMSEQ_unique)
```
```   538 done
```
```   539
```
```   540 text{*The universal quantifiers below are required for the declaration
```
```   541   of @{text Bolzano_nest_unique} below.*}
```
```   542
```
```   543 lemma Bolzano_bisect_le:
```
```   544  "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
```
```   545 apply (rule allI)
```
```   546 apply (induct_tac "n")
```
```   547 apply (auto simp add: Let_def split_def)
```
```   548 done
```
```   549
```
```   550 lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
```
```   551    \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
```
```   552 apply (rule allI)
```
```   553 apply (induct_tac "n")
```
```   554 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
```
```   555 done
```
```   556
```
```   557 lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
```
```   558    \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
```
```   559 apply (rule allI)
```
```   560 apply (induct_tac "n")
```
```   561 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
```
```   562 done
```
```   563
```
```   564 lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
```
```   565 apply (auto)
```
```   566 apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
```
```   567 apply (simp)
```
```   568 done
```
```   569
```
```   570 lemma Bolzano_bisect_diff:
```
```   571      "a \<le> b ==>
```
```   572       snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
```
```   573       (b-a) / (2 ^ n)"
```
```   574 apply (induct "n")
```
```   575 apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
```
```   576 done
```
```   577
```
```   578 lemmas Bolzano_nest_unique =
```
```   579     lemma_nest_unique
```
```   580     [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
```
```   581
```
```   582
```
```   583 lemma not_P_Bolzano_bisect:
```
```   584   assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
```
```   585       and notP: "~ P(a,b)"
```
```   586       and le:   "a \<le> b"
```
```   587   shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
```
```   588 proof (induct n)
```
```   589   case 0 show ?case using notP by simp
```
```   590  next
```
```   591   case (Suc n)
```
```   592   thus ?case
```
```   593  by (auto simp del: surjective_pairing [symmetric]
```
```   594              simp add: Let_def split_def Bolzano_bisect_le [OF le]
```
```   595      P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
```
```   596 qed
```
```   597
```
```   598 (*Now we re-package P_prem as a formula*)
```
```   599 lemma not_P_Bolzano_bisect':
```
```   600      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
```
```   601          ~ P(a,b);  a \<le> b |] ==>
```
```   602       \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
```
```   603 by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
```
```   604
```
```   605
```
```   606
```
```   607 lemma lemma_BOLZANO:
```
```   608      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> 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          a \<le> b |]
```
```   612       ==> P(a,b)"
```
```   613 apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
```
```   614 apply (rule LIMSEQ_minus_cancel)
```
```   615 apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
```
```   616 apply (rule ccontr)
```
```   617 apply (drule not_P_Bolzano_bisect', assumption+)
```
```   618 apply (rename_tac "l")
```
```   619 apply (drule_tac x = l in spec, clarify)
```
```   620 apply (simp add: LIMSEQ_iff)
```
```   621 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
```
```   622 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
```
```   623 apply (drule real_less_half_sum, auto)
```
```   624 apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
```
```   625 apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
```
```   626 apply safe
```
```   627 apply (simp_all (no_asm_simp))
```
```   628 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)
```
```   629 apply (simp (no_asm_simp) add: abs_if)
```
```   630 apply (rule real_sum_of_halves [THEN subst])
```
```   631 apply (rule add_strict_mono)
```
```   632 apply (simp_all add: diff_minus [symmetric])
```
```   633 done
```
```   634
```
```   635
```
```   636 lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
```
```   637        (\<forall>x. \<exists>d::real. 0 < d &
```
```   638                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
```
```   639       --> (\<forall>a b. a \<le> b --> P(a,b))"
```
```   640 apply clarify
```
```   641 apply (blast intro: lemma_BOLZANO)
```
```   642 done
```
```   643
```
```   644
```
```   645 subsection {* Intermediate Value Theorem *}
```
```   646
```
```   647 text {*Prove Contrapositive by Bisection*}
```
```   648
```
```   649 lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
```
```   650          a \<le> b;
```
```   651          (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
```
```   652       ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
```
```   653 apply (rule contrapos_pp, assumption)
```
```   654 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)
```
```   655 apply safe
```
```   656 apply simp_all
```
```   657 apply (simp add: isCont_iff LIM_eq)
```
```   658 apply (rule ccontr)
```
```   659 apply (subgoal_tac "a \<le> x & x \<le> b")
```
```   660  prefer 2
```
```   661  apply simp
```
```   662  apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
```
```   663 apply (drule_tac x = x in spec)+
```
```   664 apply simp
```
```   665 apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
```
```   666 apply safe
```
```   667 apply simp
```
```   668 apply (drule_tac x = s in spec, clarify)
```
```   669 apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
```
```   670 apply (drule_tac x = "ba-x" in spec)
```
```   671 apply (simp_all add: abs_if)
```
```   672 apply (drule_tac x = "aa-x" in spec)
```
```   673 apply (case_tac "x \<le> aa", simp_all)
```
```   674 done
```
```   675
```
```   676 lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
```
```   677          a \<le> b;
```
```   678          (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
```
```   679       |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
```
```   680 apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
```
```   681 apply (drule IVT [where f = "%x. - f x"], assumption)
```
```   682 apply (auto intro: isCont_minus)
```
```   683 done
```
```   684
```
```   685 (*HOL style here: object-level formulations*)
```
```   686 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
```
```   687       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```   688       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```   689 apply (blast intro: IVT)
```
```   690 done
```
```   691
```
```   692 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
```
```   693       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```   694       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```   695 apply (blast intro: IVT2)
```
```   696 done
```
```   697
```
```   698
```
```   699 subsection {* Boundedness of continuous functions *}
```
```   700
```
```   701 text{*By bisection, function continuous on closed interval is bounded above*}
```
```   702
```
```   703 lemma isCont_bounded:
```
```   704      "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   705       ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
```
```   706 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)
```
```   707 apply safe
```
```   708 apply simp_all
```
```   709 apply (rename_tac x xa ya M Ma)
```
```   710 apply (metis linorder_not_less order_le_less real_le_trans)
```
```   711 apply (case_tac "a \<le> x & x \<le> b")
```
```   712  prefer 2
```
```   713  apply (rule_tac x = 1 in exI, force)
```
```   714 apply (simp add: LIM_eq isCont_iff)
```
```   715 apply (drule_tac x = x in spec, auto)
```
```   716 apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
```
```   717 apply (drule_tac x = 1 in spec, auto)
```
```   718 apply (rule_tac x = s in exI, clarify)
```
```   719 apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
```
```   720 apply (drule_tac x = "xa-x" in spec)
```
```   721 apply (auto simp add: abs_ge_self)
```
```   722 done
```
```   723
```
```   724 text{*Refine the above to existence of least upper bound*}
```
```   725
```
```   726 lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
```
```   727       (\<exists>t. isLub UNIV S t)"
```
```   728 by (blast intro: reals_complete)
```
```   729
```
```   730 lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   731          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
```
```   732                    (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
```
```   733 apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
```
```   734         in lemma_reals_complete)
```
```   735 apply auto
```
```   736 apply (drule isCont_bounded, assumption)
```
```   737 apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
```
```   738 apply (rule exI, auto)
```
```   739 apply (auto dest!: spec simp add: linorder_not_less)
```
```   740 done
```
```   741
```
```   742 text{*Now show that it attains its upper bound*}
```
```   743
```
```   744 lemma isCont_eq_Ub:
```
```   745   assumes le: "a \<le> b"
```
```   746       and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
```
```   747   shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
```
```   748              (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
```
```   749 proof -
```
```   750   from isCont_has_Ub [OF le con]
```
```   751   obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
```
```   752              and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
```
```   753   show ?thesis
```
```   754   proof (intro exI, intro conjI)
```
```   755     show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
```
```   756     show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
```
```   757     proof (rule ccontr)
```
```   758       assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
```
```   759       with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
```
```   760         by (fastsimp simp add: linorder_not_le [symmetric])
```
```   761       hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
```
```   762         by (auto simp add: isCont_inverse isCont_diff con)
```
```   763       from isCont_bounded [OF le this]
```
```   764       obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
```
```   765       have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
```
```   766         by (simp add: M3 algebra_simps)
```
```   767       have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
```
```   768         by (auto intro: order_le_less_trans [of _ k])
```
```   769       with Minv
```
```   770       have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
```
```   771         by (intro strip less_imp_inverse_less, simp_all)
```
```   772       hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
```
```   773         by simp
```
```   774       have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
```
```   775         by (simp, arith)
```
```   776       from M2 [OF this]
```
```   777       obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
```
```   778       thus False using invlt [of x] by force
```
```   779     qed
```
```   780   qed
```
```   781 qed
```
```   782
```
```   783
```
```   784 text{*Same theorem for lower bound*}
```
```   785
```
```   786 lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   787          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
```
```   788                    (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
```
```   789 apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
```
```   790 prefer 2 apply (blast intro: isCont_minus)
```
```   791 apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
```
```   792 apply safe
```
```   793 apply auto
```
```   794 done
```
```   795
```
```   796
```
```   797 text{*Another version.*}
```
```   798
```
```   799 lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   800       ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
```
```   801           (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
```
```   802 apply (frule isCont_eq_Lb)
```
```   803 apply (frule_tac [2] isCont_eq_Ub)
```
```   804 apply (assumption+, safe)
```
```   805 apply (rule_tac x = "f x" in exI)
```
```   806 apply (rule_tac x = "f xa" in exI, simp, safe)
```
```   807 apply (cut_tac x = x and y = xa in linorder_linear, safe)
```
```   808 apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
```
```   809 apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
```
```   810 apply (rule_tac [2] x = xb in exI)
```
```   811 apply (rule_tac [4] x = xb in exI, simp_all)
```
```   812 done
```
```   813
```
```   814
```
```   815 subsection {* Local extrema *}
```
```   816
```
```   817 text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
```
```   818
```
```   819 lemma DERIV_pos_inc_right:
```
```   820   fixes f :: "real => real"
```
```   821   assumes der: "DERIV f x :> l"
```
```   822       and l:   "0 < l"
```
```   823   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
```
```   824 proof -
```
```   825   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
```
```   826   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)"
```
```   827     by (simp add: diff_minus)
```
```   828   then obtain s
```
```   829         where s:   "0 < s"
```
```   830           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
```
```   831     by auto
```
```   832   thus ?thesis
```
```   833   proof (intro exI conjI strip)
```
```   834     show "0<s" using s .
```
```   835     fix h::real
```
```   836     assume "0 < h" "h < s"
```
```   837     with all [of h] show "f x < f (x+h)"
```
```   838     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   839     split add: split_if_asm)
```
```   840       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
```
```   841       with l
```
```   842       have "0 < (f (x+h) - f x) / h" by arith
```
```   843       thus "f x < f (x+h)"
```
```   844   by (simp add: pos_less_divide_eq h)
```
```   845     qed
```
```   846   qed
```
```   847 qed
```
```   848
```
```   849 lemma DERIV_neg_dec_left:
```
```   850   fixes f :: "real => real"
```
```   851   assumes der: "DERIV f x :> l"
```
```   852       and l:   "l < 0"
```
```   853   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
```
```   854 proof -
```
```   855   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
```
```   856   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)"
```
```   857     by (simp add: diff_minus)
```
```   858   then obtain s
```
```   859         where s:   "0 < s"
```
```   860           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
```
```   861     by auto
```
```   862   thus ?thesis
```
```   863   proof (intro exI conjI strip)
```
```   864     show "0<s" using s .
```
```   865     fix h::real
```
```   866     assume "0 < h" "h < s"
```
```   867     with all [of "-h"] show "f x < f (x-h)"
```
```   868     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   869     split add: split_if_asm)
```
```   870       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
```
```   871       with l
```
```   872       have "0 < (f (x-h) - f x) / h" by arith
```
```   873       thus "f x < f (x-h)"
```
```   874   by (simp add: pos_less_divide_eq h)
```
```   875     qed
```
```   876   qed
```
```   877 qed
```
```   878
```
```   879
```
```   880 lemma DERIV_pos_inc_left:
```
```   881   fixes f :: "real => real"
```
```   882   shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
```
```   883   apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])
```
```   884   apply (auto simp add: DERIV_minus)
```
```   885   done
```
```   886
```
```   887 lemma DERIV_neg_dec_right:
```
```   888   fixes f :: "real => real"
```
```   889   shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
```
```   890   apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])
```
```   891   apply (auto simp add: DERIV_minus)
```
```   892   done
```
```   893
```
```   894 lemma DERIV_local_max:
```
```   895   fixes f :: "real => real"
```
```   896   assumes der: "DERIV f x :> l"
```
```   897       and d:   "0 < d"
```
```   898       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
```
```   899   shows "l = 0"
```
```   900 proof (cases rule: linorder_cases [of l 0])
```
```   901   case equal thus ?thesis .
```
```   902 next
```
```   903   case less
```
```   904   from DERIV_neg_dec_left [OF der less]
```
```   905   obtain d' where d': "0 < d'"
```
```   906              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
```
```   907   from real_lbound_gt_zero [OF d d']
```
```   908   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   909   with lt le [THEN spec [where x="x-e"]]
```
```   910   show ?thesis by (auto simp add: abs_if)
```
```   911 next
```
```   912   case greater
```
```   913   from DERIV_pos_inc_right [OF der greater]
```
```   914   obtain d' where d': "0 < d'"
```
```   915              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
```
```   916   from real_lbound_gt_zero [OF d d']
```
```   917   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   918   with lt le [THEN spec [where x="x+e"]]
```
```   919   show ?thesis by (auto simp add: abs_if)
```
```   920 qed
```
```   921
```
```   922
```
```   923 text{*Similar theorem for a local minimum*}
```
```   924 lemma DERIV_local_min:
```
```   925   fixes f :: "real => real"
```
```   926   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
```
```   927 by (drule DERIV_minus [THEN DERIV_local_max], auto)
```
```   928
```
```   929
```
```   930 text{*In particular, if a function is locally flat*}
```
```   931 lemma DERIV_local_const:
```
```   932   fixes f :: "real => real"
```
```   933   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
```
```   934 by (auto dest!: DERIV_local_max)
```
```   935
```
```   936
```
```   937 subsection {* Rolle's Theorem *}
```
```   938
```
```   939 text{*Lemma about introducing open ball in open interval*}
```
```   940 lemma lemma_interval_lt:
```
```   941      "[| a < x;  x < b |]
```
```   942       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
```
```   943
```
```   944 apply (simp add: abs_less_iff)
```
```   945 apply (insert linorder_linear [of "x-a" "b-x"], safe)
```
```   946 apply (rule_tac x = "x-a" in exI)
```
```   947 apply (rule_tac [2] x = "b-x" in exI, auto)
```
```   948 done
```
```   949
```
```   950 lemma lemma_interval: "[| a < x;  x < b |] ==>
```
```   951         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
```
```   952 apply (drule lemma_interval_lt, auto)
```
```   953 apply (auto intro!: exI)
```
```   954 done
```
```   955
```
```   956 text{*Rolle's Theorem.
```
```   957    If @{term f} is defined and continuous on the closed interval
```
```   958    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
```
```   959    and @{term "f(a) = f(b)"},
```
```   960    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
```
```   961 theorem Rolle:
```
```   962   assumes lt: "a < b"
```
```   963       and eq: "f(a) = f(b)"
```
```   964       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```   965       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```   966   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
```
```   967 proof -
```
```   968   have le: "a \<le> b" using lt by simp
```
```   969   from isCont_eq_Ub [OF le con]
```
```   970   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
```
```   971              and alex: "a \<le> x" and xleb: "x \<le> b"
```
```   972     by blast
```
```   973   from isCont_eq_Lb [OF le con]
```
```   974   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
```
```   975               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
```
```   976     by blast
```
```   977   show ?thesis
```
```   978   proof cases
```
```   979     assume axb: "a < x & x < b"
```
```   980         --{*@{term f} attains its maximum within the interval*}
```
```   981     hence ax: "a<x" and xb: "x<b" by arith +
```
```   982     from lemma_interval [OF ax xb]
```
```   983     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   984       by blast
```
```   985     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
```
```   986       by blast
```
```   987     from differentiableD [OF dif [OF axb]]
```
```   988     obtain l where der: "DERIV f x :> l" ..
```
```   989     have "l=0" by (rule DERIV_local_max [OF der d bound'])
```
```   990         --{*the derivative at a local maximum is zero*}
```
```   991     thus ?thesis using ax xb der by auto
```
```   992   next
```
```   993     assume notaxb: "~ (a < x & x < b)"
```
```   994     hence xeqab: "x=a | x=b" using alex xleb by arith
```
```   995     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
```
```   996     show ?thesis
```
```   997     proof cases
```
```   998       assume ax'b: "a < x' & x' < b"
```
```   999         --{*@{term f} attains its minimum within the interval*}
```
```  1000       hence ax': "a<x'" and x'b: "x'<b" by arith+
```
```  1001       from lemma_interval [OF ax' x'b]
```
```  1002       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```  1003   by blast
```
```  1004       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
```
```  1005   by blast
```
```  1006       from differentiableD [OF dif [OF ax'b]]
```
```  1007       obtain l where der: "DERIV f x' :> l" ..
```
```  1008       have "l=0" by (rule DERIV_local_min [OF der d bound'])
```
```  1009         --{*the derivative at a local minimum is zero*}
```
```  1010       thus ?thesis using ax' x'b der by auto
```
```  1011     next
```
```  1012       assume notax'b: "~ (a < x' & x' < b)"
```
```  1013         --{*@{term f} is constant througout the interval*}
```
```  1014       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
```
```  1015       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
```
```  1016       from dense [OF lt]
```
```  1017       obtain r where ar: "a < r" and rb: "r < b" by blast
```
```  1018       from lemma_interval [OF ar rb]
```
```  1019       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```  1020   by blast
```
```  1021       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
```
```  1022       proof (clarify)
```
```  1023         fix z::real
```
```  1024         assume az: "a \<le> z" and zb: "z \<le> b"
```
```  1025         show "f z = f b"
```
```  1026         proof (rule order_antisym)
```
```  1027           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
```
```  1028           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
```
```  1029         qed
```
```  1030       qed
```
```  1031       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
```
```  1032       proof (intro strip)
```
```  1033         fix y::real
```
```  1034         assume lt: "\<bar>r-y\<bar> < d"
```
```  1035         hence "f y = f b" by (simp add: eq_fb bound)
```
```  1036         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
```
```  1037       qed
```
```  1038       from differentiableD [OF dif [OF conjI [OF ar rb]]]
```
```  1039       obtain l where der: "DERIV f r :> l" ..
```
```  1040       have "l=0" by (rule DERIV_local_const [OF der d bound'])
```
```  1041         --{*the derivative of a constant function is zero*}
```
```  1042       thus ?thesis using ar rb der by auto
```
```  1043     qed
```
```  1044   qed
```
```  1045 qed
```
```  1046
```
```  1047
```
```  1048 subsection{*Mean Value Theorem*}
```
```  1049
```
```  1050 lemma lemma_MVT:
```
```  1051      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
```
```  1052 proof cases
```
```  1053   assume "a=b" thus ?thesis by simp
```
```  1054 next
```
```  1055   assume "a\<noteq>b"
```
```  1056   hence ba: "b-a \<noteq> 0" by arith
```
```  1057   show ?thesis
```
```  1058     by (rule real_mult_left_cancel [OF ba, THEN iffD1],
```
```  1059         simp add: right_diff_distrib,
```
```  1060         simp add: left_diff_distrib)
```
```  1061 qed
```
```  1062
```
```  1063 theorem MVT:
```
```  1064   assumes lt:  "a < b"
```
```  1065       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```  1066       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```  1067   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
```
```  1068                    (f(b) - f(a) = (b-a) * l)"
```
```  1069 proof -
```
```  1070   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
```
```  1071   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
```
```  1072     by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
```
```  1073   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
```
```  1074   proof (clarify)
```
```  1075     fix x::real
```
```  1076     assume ax: "a < x" and xb: "x < b"
```
```  1077     from differentiableD [OF dif [OF conjI [OF ax xb]]]
```
```  1078     obtain l where der: "DERIV f x :> l" ..
```
```  1079     show "?F differentiable x"
```
```  1080       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
```
```  1081           blast intro: DERIV_diff DERIV_cmult_Id der)
```
```  1082   qed
```
```  1083   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
```
```  1084   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
```
```  1085     by blast
```
```  1086   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
```
```  1087     by (rule DERIV_cmult_Id)
```
```  1088   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
```
```  1089                    :> 0 + (f b - f a) / (b - a)"
```
```  1090     by (rule DERIV_add [OF der])
```
```  1091   show ?thesis
```
```  1092   proof (intro exI conjI)
```
```  1093     show "a < z" using az .
```
```  1094     show "z < b" using zb .
```
```  1095     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
```
```  1096     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
```
```  1097   qed
```
```  1098 qed
```
```  1099
```
```  1100 lemma MVT2:
```
```  1101      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
```
```  1102       ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
```
```  1103 apply (drule MVT)
```
```  1104 apply (blast intro: DERIV_isCont)
```
```  1105 apply (force dest: order_less_imp_le simp add: differentiable_def)
```
```  1106 apply (blast dest: DERIV_unique order_less_imp_le)
```
```  1107 done
```
```  1108
```
```  1109
```
```  1110 text{*A function is constant if its derivative is 0 over an interval.*}
```
```  1111
```
```  1112 lemma DERIV_isconst_end:
```
```  1113   fixes f :: "real => real"
```
```  1114   shows "[| a < b;
```
```  1115          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1116          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1117         ==> f b = f a"
```
```  1118 apply (drule MVT, assumption)
```
```  1119 apply (blast intro: differentiableI)
```
```  1120 apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
```
```  1121 done
```
```  1122
```
```  1123 lemma DERIV_isconst1:
```
```  1124   fixes f :: "real => real"
```
```  1125   shows "[| a < b;
```
```  1126          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1127          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1128         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
```
```  1129 apply safe
```
```  1130 apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
```
```  1131 apply (drule_tac b = x in DERIV_isconst_end, auto)
```
```  1132 done
```
```  1133
```
```  1134 lemma DERIV_isconst2:
```
```  1135   fixes f :: "real => real"
```
```  1136   shows "[| a < b;
```
```  1137          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1138          \<forall>x. a < x & x < b --> DERIV f x :> 0;
```
```  1139          a \<le> x; x \<le> b |]
```
```  1140         ==> f x = f a"
```
```  1141 apply (blast dest: DERIV_isconst1)
```
```  1142 done
```
```  1143
```
```  1144 lemma DERIV_isconst3: fixes a b x y :: real
```
```  1145   assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
```
```  1146   assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
```
```  1147   shows "f x = f y"
```
```  1148 proof (cases "x = y")
```
```  1149   case False
```
```  1150   let ?a = "min x y"
```
```  1151   let ?b = "max x y"
```
```  1152
```
```  1153   have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
```
```  1154   proof (rule allI, rule impI)
```
```  1155     fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
```
```  1156     hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
```
```  1157     hence "z \<in> {a<..<b}" by auto
```
```  1158     thus "DERIV f z :> 0" by (rule derivable)
```
```  1159   qed
```
```  1160   hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
```
```  1161     and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
```
```  1162
```
```  1163   have "?a < ?b" using `x \<noteq> y` by auto
```
```  1164   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
```
```  1165   show ?thesis by auto
```
```  1166 qed auto
```
```  1167
```
```  1168 lemma DERIV_isconst_all:
```
```  1169   fixes f :: "real => real"
```
```  1170   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
```
```  1171 apply (rule linorder_cases [of x y])
```
```  1172 apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
```
```  1173 done
```
```  1174
```
```  1175 lemma DERIV_const_ratio_const:
```
```  1176   fixes f :: "real => real"
```
```  1177   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
```
```  1178 apply (rule linorder_cases [of a b], auto)
```
```  1179 apply (drule_tac [!] f = f in MVT)
```
```  1180 apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
```
```  1181 apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
```
```  1182 done
```
```  1183
```
```  1184 lemma DERIV_const_ratio_const2:
```
```  1185   fixes f :: "real => real"
```
```  1186   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
```
```  1187 apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
```
```  1188 apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
```
```  1189 done
```
```  1190
```
```  1191 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
```
```  1192 by (simp)
```
```  1193
```
```  1194 lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
```
```  1195 by (simp)
```
```  1196
```
```  1197 text{*Gallileo's "trick": average velocity = av. of end velocities*}
```
```  1198
```
```  1199 lemma DERIV_const_average:
```
```  1200   fixes v :: "real => real"
```
```  1201   assumes neq: "a \<noteq> (b::real)"
```
```  1202       and der: "\<forall>x. DERIV v x :> k"
```
```  1203   shows "v ((a + b)/2) = (v a + v b)/2"
```
```  1204 proof (cases rule: linorder_cases [of a b])
```
```  1205   case equal with neq show ?thesis by simp
```
```  1206 next
```
```  1207   case less
```
```  1208   have "(v b - v a) / (b - a) = k"
```
```  1209     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1210   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1211   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
```
```  1212     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1213   ultimately show ?thesis using neq by force
```
```  1214 next
```
```  1215   case greater
```
```  1216   have "(v b - v a) / (b - a) = k"
```
```  1217     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1218   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1219   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
```
```  1220     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1221   ultimately show ?thesis using neq by (force simp add: add_commute)
```
```  1222 qed
```
```  1223
```
```  1224 (* A function with positive derivative is increasing.
```
```  1225    A simple proof using the MVT, by Jeremy Avigad. And variants.
```
```  1226 *)
```
```  1227
```
```  1228 lemma DERIV_pos_imp_increasing:
```
```  1229   fixes a::real and b::real and f::"real => real"
```
```  1230   assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
```
```  1231   shows "f a < f b"
```
```  1232 proof (rule ccontr)
```
```  1233   assume "~ f a < f b"
```
```  1234   from assms have "EX l z. a < z & z < b & DERIV f z :> l
```
```  1235       & f b - f a = (b - a) * l"
```
```  1236     by (metis MVT DERIV_isCont differentiableI real_less_def)
```
```  1237   then obtain l z where "a < z" and "z < b" and "DERIV f z :> l"
```
```  1238       and "f b - f a = (b - a) * l"
```
```  1239     by auto
```
```  1240
```
```  1241   from prems have "~(l > 0)"
```
```  1242     by (metis assms(1) linorder_not_le mult_le_0_iff real_le_eq_diff)
```
```  1243   with prems show False
```
```  1244     by (metis DERIV_unique real_less_def)
```
```  1245 qed
```
```  1246
```
```  1247
```
```  1248 lemma DERIV_nonneg_imp_nonincreasing:
```
```  1249   fixes a::real and b::real and f::"real => real"
```
```  1250   assumes "a \<le> b" and
```
```  1251     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
```
```  1252   shows "f a \<le> f b"
```
```  1253 proof (rule ccontr, cases "a = b")
```
```  1254   assume "~ f a \<le> f b"
```
```  1255   assume "a = b"
```
```  1256   with prems show False by auto
```
```  1257   next assume "~ f a \<le> f b"
```
```  1258   assume "a ~= b"
```
```  1259   with assms have "EX l z. a < z & z < b & DERIV f z :> l
```
```  1260       & f b - f a = (b - a) * l"
```
```  1261     apply (intro MVT)
```
```  1262     apply auto
```
```  1263     apply (metis DERIV_isCont)
```
```  1264     apply (metis differentiableI real_less_def)
```
```  1265     done
```
```  1266   then obtain l z where "a < z" and "z < b" and "DERIV f z :> l"
```
```  1267       and "f b - f a = (b - a) * l"
```
```  1268     by auto
```
```  1269   from prems have "~(l >= 0)"
```
```  1270     by (metis diff_self le_eqI le_iff_diff_le_0 real_le_anti_sym real_le_linear
```
```  1271               split_mult_pos_le)
```
```  1272   with prems show False
```
```  1273     by (metis DERIV_unique order_less_imp_le)
```
```  1274 qed
```
```  1275
```
```  1276 lemma DERIV_neg_imp_decreasing:
```
```  1277   fixes a::real and b::real and f::"real => real"
```
```  1278   assumes "a < b" and
```
```  1279     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
```
```  1280   shows "f a > f b"
```
```  1281 proof -
```
```  1282   have "(%x. -f x) a < (%x. -f x) b"
```
```  1283     apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
```
```  1284     apply (insert prems, auto)
```
```  1285     apply (metis DERIV_minus neg_0_less_iff_less)
```
```  1286     done
```
```  1287   thus ?thesis
```
```  1288     by simp
```
```  1289 qed
```
```  1290
```
```  1291 lemma DERIV_nonpos_imp_nonincreasing:
```
```  1292   fixes a::real and b::real and f::"real => real"
```
```  1293   assumes "a \<le> b" and
```
```  1294     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
```
```  1295   shows "f a \<ge> f b"
```
```  1296 proof -
```
```  1297   have "(%x. -f x) a \<le> (%x. -f x) b"
```
```  1298     apply (rule DERIV_nonneg_imp_nonincreasing [of a b "%x. -f x"])
```
```  1299     apply (insert prems, auto)
```
```  1300     apply (metis DERIV_minus neg_0_le_iff_le)
```
```  1301     done
```
```  1302   thus ?thesis
```
```  1303     by simp
```
```  1304 qed
```
```  1305
```
```  1306 subsection {* Continuous injective functions *}
```
```  1307
```
```  1308 text{*Dull lemma: an continuous injection on an interval must have a
```
```  1309 strict maximum at an end point, not in the middle.*}
```
```  1310
```
```  1311 lemma lemma_isCont_inj:
```
```  1312   fixes f :: "real \<Rightarrow> real"
```
```  1313   assumes d: "0 < d"
```
```  1314       and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1315       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1316   shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
```
```  1317 proof (rule ccontr)
```
```  1318   assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
```
```  1319   hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
```
```  1320   show False
```
```  1321   proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
```
```  1322     case le
```
```  1323     from d cont all [of "x+d"]
```
```  1324     have flef: "f(x+d) \<le> f x"
```
```  1325      and xlex: "x - d \<le> x"
```
```  1326      and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
```
```  1327        by (auto simp add: abs_if)
```
```  1328     from IVT [OF le flef xlex cont']
```
```  1329     obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
```
```  1330     moreover
```
```  1331     hence "g(f x') = g (f(x+d))" by simp
```
```  1332     ultimately show False using d inj [of x'] inj [of "x+d"]
```
```  1333       by (simp add: abs_le_iff)
```
```  1334   next
```
```  1335     case ge
```
```  1336     from d cont all [of "x-d"]
```
```  1337     have flef: "f(x-d) \<le> f x"
```
```  1338      and xlex: "x \<le> x+d"
```
```  1339      and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
```
```  1340        by (auto simp add: abs_if)
```
```  1341     from IVT2 [OF ge flef xlex cont']
```
```  1342     obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
```
```  1343     moreover
```
```  1344     hence "g(f x') = g (f(x-d))" by simp
```
```  1345     ultimately show False using d inj [of x'] inj [of "x-d"]
```
```  1346       by (simp add: abs_le_iff)
```
```  1347   qed
```
```  1348 qed
```
```  1349
```
```  1350
```
```  1351 text{*Similar version for lower bound.*}
```
```  1352
```
```  1353 lemma lemma_isCont_inj2:
```
```  1354   fixes f g :: "real \<Rightarrow> real"
```
```  1355   shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
```
```  1356         \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
```
```  1357       ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
```
```  1358 apply (insert lemma_isCont_inj
```
```  1359           [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
```
```  1360 apply (simp add: isCont_minus linorder_not_le)
```
```  1361 done
```
```  1362
```
```  1363 text{*Show there's an interval surrounding @{term "f(x)"} in
```
```  1364 @{text "f[[x - d, x + d]]"} .*}
```
```  1365
```
```  1366 lemma isCont_inj_range:
```
```  1367   fixes f :: "real \<Rightarrow> real"
```
```  1368   assumes d: "0 < d"
```
```  1369       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1370       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1371   shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
```
```  1372 proof -
```
```  1373   have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
```
```  1374     by (auto simp add: abs_le_iff)
```
```  1375   from isCont_Lb_Ub [OF this]
```
```  1376   obtain L M
```
```  1377   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"
```
```  1378     and all2 [rule_format]:
```
```  1379            "\<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)"
```
```  1380     by auto
```
```  1381   with d have "L \<le> f x & f x \<le> M" by simp
```
```  1382   moreover have "L \<noteq> f x"
```
```  1383   proof -
```
```  1384     from lemma_isCont_inj2 [OF d inj cont]
```
```  1385     obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
```
```  1386     thus ?thesis using all1 [of u] by arith
```
```  1387   qed
```
```  1388   moreover have "f x \<noteq> M"
```
```  1389   proof -
```
```  1390     from lemma_isCont_inj [OF d inj cont]
```
```  1391     obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
```
```  1392     thus ?thesis using all1 [of u] by arith
```
```  1393   qed
```
```  1394   ultimately have "L < f x & f x < M" by arith
```
```  1395   hence "0 < f x - L" "0 < M - f x" by arith+
```
```  1396   from real_lbound_gt_zero [OF this]
```
```  1397   obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
```
```  1398   thus ?thesis
```
```  1399   proof (intro exI conjI)
```
```  1400     show "0<e" using e(1) .
```
```  1401     show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
```
```  1402     proof (intro strip)
```
```  1403       fix y::real
```
```  1404       assume "\<bar>y - f x\<bar> \<le> e"
```
```  1405       with e have "L \<le> y \<and> y \<le> M" by arith
```
```  1406       from all2 [OF this]
```
```  1407       obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
```
```  1408       thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
```
```  1409         by (force simp add: abs_le_iff)
```
```  1410     qed
```
```  1411   qed
```
```  1412 qed
```
```  1413
```
```  1414
```
```  1415 text{*Continuity of inverse function*}
```
```  1416
```
```  1417 lemma isCont_inverse_function:
```
```  1418   fixes f g :: "real \<Rightarrow> real"
```
```  1419   assumes d: "0 < d"
```
```  1420       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1421       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1422   shows "isCont g (f x)"
```
```  1423 proof (simp add: isCont_iff LIM_eq)
```
```  1424   show "\<forall>r. 0 < r \<longrightarrow>
```
```  1425          (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
```
```  1426   proof (intro strip)
```
```  1427     fix r::real
```
```  1428     assume r: "0<r"
```
```  1429     from real_lbound_gt_zero [OF r d]
```
```  1430     obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
```
```  1431     with inj cont
```
```  1432     have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
```
```  1433                   "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
```
```  1434     from isCont_inj_range [OF e this]
```
```  1435     obtain e' where e': "0 < e'"
```
```  1436         and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
```
```  1437           by blast
```
```  1438     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"
```
```  1439     proof (intro exI conjI)
```
```  1440       show "0<e'" using e' .
```
```  1441       show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
```
```  1442       proof (intro strip)
```
```  1443         fix z::real
```
```  1444         assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
```
```  1445         with e e_lt e_simps all [rule_format, of "f x + z"]
```
```  1446         show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
```
```  1447       qed
```
```  1448     qed
```
```  1449   qed
```
```  1450 qed
```
```  1451
```
```  1452 text {* Derivative of inverse function *}
```
```  1453
```
```  1454 lemma DERIV_inverse_function:
```
```  1455   fixes f g :: "real \<Rightarrow> real"
```
```  1456   assumes der: "DERIV f (g x) :> D"
```
```  1457   assumes neq: "D \<noteq> 0"
```
```  1458   assumes a: "a < x" and b: "x < b"
```
```  1459   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
```
```  1460   assumes cont: "isCont g x"
```
```  1461   shows "DERIV g x :> inverse D"
```
```  1462 unfolding DERIV_iff2
```
```  1463 proof (rule LIM_equal2)
```
```  1464   show "0 < min (x - a) (b - x)"
```
```  1465     using a b by arith
```
```  1466 next
```
```  1467   fix y
```
```  1468   assume "norm (y - x) < min (x - a) (b - x)"
```
```  1469   hence "a < y" and "y < b"
```
```  1470     by (simp_all add: abs_less_iff)
```
```  1471   thus "(g y - g x) / (y - x) =
```
```  1472         inverse ((f (g y) - x) / (g y - g x))"
```
```  1473     by (simp add: inj)
```
```  1474 next
```
```  1475   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
```
```  1476     by (rule der [unfolded DERIV_iff2])
```
```  1477   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
```
```  1478     using inj a b by simp
```
```  1479   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
```
```  1480   proof (safe intro!: exI)
```
```  1481     show "0 < min (x - a) (b - x)"
```
```  1482       using a b by simp
```
```  1483   next
```
```  1484     fix y
```
```  1485     assume "norm (y - x) < min (x - a) (b - x)"
```
```  1486     hence y: "a < y" "y < b"
```
```  1487       by (simp_all add: abs_less_iff)
```
```  1488     assume "g y = g x"
```
```  1489     hence "f (g y) = f (g x)" by simp
```
```  1490     hence "y = x" using inj y a b by simp
```
```  1491     also assume "y \<noteq> x"
```
```  1492     finally show False by simp
```
```  1493   qed
```
```  1494   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
```
```  1495     using cont 1 2 by (rule isCont_LIM_compose2)
```
```  1496   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
```
```  1497         -- x --> inverse D"
```
```  1498     using neq by (rule LIM_inverse)
```
```  1499 qed
```
```  1500
```
```  1501
```
```  1502 subsection {* Generalized Mean Value Theorem *}
```
```  1503
```
```  1504 theorem GMVT:
```
```  1505   fixes a b :: real
```
```  1506   assumes alb: "a < b"
```
```  1507   and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  1508   and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
```
```  1509   and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
```
```  1510   and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
```
```  1511   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)"
```
```  1512 proof -
```
```  1513   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
```
```  1514   from prems have "a < b" by simp
```
```  1515   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
```
```  1516   proof -
```
```  1517     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
```
```  1518     with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
```
```  1519       by (auto intro: isCont_mult)
```
```  1520     moreover
```
```  1521     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
```
```  1522     with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
```
```  1523       by (auto intro: isCont_mult)
```
```  1524     ultimately show ?thesis
```
```  1525       by (fastsimp intro: isCont_diff)
```
```  1526   qed
```
```  1527   moreover
```
```  1528   have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
```
```  1529   proof -
```
```  1530     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
```
```  1531     with gd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (f b - f a) * g x) differentiable x" by (simp add: differentiable_mult)
```
```  1532     moreover
```
```  1533     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
```
```  1534     with fd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (g b - g a) * f x) differentiable x" by (simp add: differentiable_mult)
```
```  1535     ultimately show ?thesis by (simp add: differentiable_diff)
```
```  1536   qed
```
```  1537   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)
```
```  1538   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" ..
```
```  1539   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
```
```  1540
```
```  1541   from cdef have cint: "a < c \<and> c < b" by auto
```
```  1542   with gd have "g differentiable c" by simp
```
```  1543   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
```
```  1544   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
```
```  1545
```
```  1546   from cdef have "a < c \<and> c < b" by auto
```
```  1547   with fd have "f differentiable c" by simp
```
```  1548   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
```
```  1549   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
```
```  1550
```
```  1551   from cdef have "DERIV ?h c :> l" by auto
```
```  1552   moreover
```
```  1553   {
```
```  1554     have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
```
```  1555       apply (insert DERIV_const [where k="f b - f a"])
```
```  1556       apply (drule meta_spec [of _ c])
```
```  1557       apply (drule DERIV_mult [OF _ g'cdef])
```
```  1558       by simp
```
```  1559     moreover have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
```
```  1560       apply (insert DERIV_const [where k="g b - g a"])
```
```  1561       apply (drule meta_spec [of _ c])
```
```  1562       apply (drule DERIV_mult [OF _ f'cdef])
```
```  1563       by simp
```
```  1564     ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
```
```  1565       by (simp add: DERIV_diff)
```
```  1566   }
```
```  1567   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
```
```  1568
```
```  1569   {
```
```  1570     from cdef have "?h b - ?h a = (b - a) * l" by auto
```
```  1571     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1572     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1573   }
```
```  1574   moreover
```
```  1575   {
```
```  1576     have "?h b - ?h a =
```
```  1577          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
```
```  1578           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
```
```  1579       by (simp add: algebra_simps)
```
```  1580     hence "?h b - ?h a = 0" by auto
```
```  1581   }
```
```  1582   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
```
```  1583   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
```
```  1584   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
```
```  1585   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
```
```  1586
```
```  1587   with g'cdef f'cdef cint show ?thesis by auto
```
```  1588 qed
```
```  1589
```
```  1590
```
```  1591 subsection {* Theorems about Limits *}
```
```  1592
```
```  1593 (* need to rename second isCont_inverse *)
```
```  1594
```
```  1595 lemma isCont_inv_fun:
```
```  1596   fixes f g :: "real \<Rightarrow> real"
```
```  1597   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  1598          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  1599       ==> isCont g (f x)"
```
```  1600 by (rule isCont_inverse_function)
```
```  1601
```
```  1602 lemma isCont_inv_fun_inv:
```
```  1603   fixes f g :: "real \<Rightarrow> real"
```
```  1604   shows "[| 0 < d;
```
```  1605          \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  1606          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  1607        ==> \<exists>e. 0 < e &
```
```  1608              (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
```
```  1609 apply (drule isCont_inj_range)
```
```  1610 prefer 2 apply (assumption, assumption, auto)
```
```  1611 apply (rule_tac x = e in exI, auto)
```
```  1612 apply (rotate_tac 2)
```
```  1613 apply (drule_tac x = y in spec, auto)
```
```  1614 done
```
```  1615
```
```  1616
```
```  1617 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
```
```  1618 lemma LIM_fun_gt_zero:
```
```  1619      "[| f -- c --> (l::real); 0 < l |]
```
```  1620          ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
```
```  1621 apply (auto simp add: LIM_eq)
```
```  1622 apply (drule_tac x = "l/2" in spec, safe, force)
```
```  1623 apply (rule_tac x = s in exI)
```
```  1624 apply (auto simp only: abs_less_iff)
```
```  1625 done
```
```  1626
```
```  1627 lemma LIM_fun_less_zero:
```
```  1628      "[| f -- c --> (l::real); l < 0 |]
```
```  1629       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
```
```  1630 apply (auto simp add: LIM_eq)
```
```  1631 apply (drule_tac x = "-l/2" in spec, safe, force)
```
```  1632 apply (rule_tac x = s in exI)
```
```  1633 apply (auto simp only: abs_less_iff)
```
```  1634 done
```
```  1635
```
```  1636
```
```  1637 lemma LIM_fun_not_zero:
```
```  1638      "[| f -- c --> (l::real); l \<noteq> 0 |]
```
```  1639       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
```
```  1640 apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
```
```  1641 apply (drule LIM_fun_less_zero)
```
```  1642 apply (drule_tac [3] LIM_fun_gt_zero)
```
```  1643 apply force+
```
```  1644 done
```
```  1645
```
```  1646 end
```