src/HOL/Hyperreal/Deriv.thy
 author chaieb Tue Jun 05 16:31:10 2007 +0200 (2007-06-05) changeset 23255 631bd424fd72 parent 23069 cdfff0241c12 child 23398 0b5a400c7595 permissions -rw-r--r--
lemma lemma_DERIV_subst moved to Deriv.thy
```     1 (*  Title       : Deriv.thy
```
```     2     ID          : \$Id\$
```
```     3     Author      : Jacques D. Fleuriot
```
```     4     Copyright   : 1998  University of Cambridge
```
```     5     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
```
```     6     GMVT by Benjamin Porter, 2005
```
```     7 *)
```
```     8
```
```     9 header{* Differentiation *}
```
```    10
```
```    11 theory Deriv
```
```    12 imports Lim
```
```    13 begin
```
```    14
```
```    15 text{*Standard Definitions*}
```
```    16
```
```    17 definition
```
```    18   deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
```
```    19     --{*Differentiation: D is derivative of function f at x*}
```
```    20           ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
```
```    21   "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"
```
```    22
```
```    23 definition
```
```    24   differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
```
```    25     (infixl "differentiable" 60) where
```
```    26   "f differentiable x = (\<exists>D. DERIV f x :> D)"
```
```    27
```
```    28
```
```    29 consts
```
```    30   Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
```
```    31 primrec
```
```    32   "Bolzano_bisect P a b 0 = (a,b)"
```
```    33   "Bolzano_bisect P a b (Suc n) =
```
```    34       (let (x,y) = Bolzano_bisect P a b n
```
```    35        in if P(x, (x+y)/2) then ((x+y)/2, y)
```
```    36                             else (x, (x+y)/2))"
```
```    37
```
```    38
```
```    39 subsection {* Derivatives *}
```
```    40
```
```    41 lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
```
```    42 by (simp add: deriv_def)
```
```    43
```
```    44 lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
```
```    45 by (simp add: deriv_def)
```
```    46
```
```    47 lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
```
```    48 by (simp add: deriv_def)
```
```    49
```
```    50 lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
```
```    51 by (simp add: deriv_def divide_self cong: LIM_cong)
```
```    52
```
```    53 lemma add_diff_add:
```
```    54   fixes a b c d :: "'a::ab_group_add"
```
```    55   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```    56 by simp
```
```    57
```
```    58 lemma DERIV_add:
```
```    59   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
```
```    60 by (simp only: deriv_def add_diff_add add_divide_distrib LIM_add)
```
```    61
```
```    62 lemma DERIV_minus:
```
```    63   "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
```
```    64 by (simp only: deriv_def minus_diff_minus divide_minus_left LIM_minus)
```
```    65
```
```    66 lemma DERIV_diff:
```
```    67   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
```
```    68 by (simp only: diff_def DERIV_add DERIV_minus)
```
```    69
```
```    70 lemma DERIV_add_minus:
```
```    71   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
```
```    72 by (simp only: DERIV_add DERIV_minus)
```
```    73
```
```    74 lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
```
```    75 proof (unfold isCont_iff)
```
```    76   assume "DERIV f x :> D"
```
```    77   hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"
```
```    78     by (rule DERIV_D)
```
```    79   hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
```
```    80     by (intro LIM_mult LIM_ident)
```
```    81   hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
```
```    82     by simp
```
```    83   hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
```
```    84     by (simp cong: LIM_cong add: divide_self)
```
```    85   thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
```
```    86     by (simp add: LIM_def)
```
```    87 qed
```
```    88
```
```    89 lemma DERIV_mult_lemma:
```
```    90   fixes a b c d :: "'a::real_field"
```
```    91   shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"
```
```    92 by (simp add: diff_minus add_divide_distrib [symmetric] ring_distrib)
```
```    93
```
```    94 lemma DERIV_mult':
```
```    95   assumes f: "DERIV f x :> D"
```
```    96   assumes g: "DERIV g x :> E"
```
```    97   shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
```
```    98 proof (unfold deriv_def)
```
```    99   from f have "isCont f x"
```
```   100     by (rule DERIV_isCont)
```
```   101   hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
```
```   102     by (simp only: isCont_iff)
```
```   103   hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +
```
```   104               ((f(x+h) - f x) / h) * g x)
```
```   105           -- 0 --> f x * E + D * g x"
```
```   106     by (intro LIM_add LIM_mult LIM_const DERIV_D f g)
```
```   107   thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)
```
```   108          -- 0 --> f x * E + D * g x"
```
```   109     by (simp only: DERIV_mult_lemma)
```
```   110 qed
```
```   111
```
```   112 lemma DERIV_mult:
```
```   113      "[| DERIV f x :> Da; DERIV g x :> Db |]
```
```   114       ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
```
```   115 by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
```
```   116
```
```   117 lemma DERIV_unique:
```
```   118       "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
```
```   119 apply (simp add: deriv_def)
```
```   120 apply (blast intro: LIM_unique)
```
```   121 done
```
```   122
```
```   123 text{*Differentiation of finite sum*}
```
```   124
```
```   125 lemma DERIV_sumr [rule_format (no_asm)]:
```
```   126      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
```
```   127       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
```
```   128 apply (induct "n")
```
```   129 apply (auto intro: DERIV_add)
```
```   130 done
```
```   131
```
```   132 text{*Alternative definition for differentiability*}
```
```   133
```
```   134 lemma DERIV_LIM_iff:
```
```   135      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
```
```   136       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
```
```   137 apply (rule iffI)
```
```   138 apply (drule_tac k="- a" in LIM_offset)
```
```   139 apply (simp add: diff_minus)
```
```   140 apply (drule_tac k="a" in LIM_offset)
```
```   141 apply (simp add: add_commute)
```
```   142 done
```
```   143
```
```   144 lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
```
```   145 by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
```
```   146
```
```   147 lemma inverse_diff_inverse:
```
```   148   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   149    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   150 by (simp add: right_diff_distrib left_diff_distrib mult_assoc)
```
```   151
```
```   152 lemma DERIV_inverse_lemma:
```
```   153   "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
```
```   154    \<Longrightarrow> (inverse a - inverse b) / h
```
```   155      = - (inverse a * ((a - b) / h) * inverse b)"
```
```   156 by (simp add: inverse_diff_inverse)
```
```   157
```
```   158 lemma DERIV_inverse':
```
```   159   assumes der: "DERIV f x :> D"
```
```   160   assumes neq: "f x \<noteq> 0"
```
```   161   shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
```
```   162     (is "DERIV _ _ :> ?E")
```
```   163 proof (unfold DERIV_iff2)
```
```   164   from der have lim_f: "f -- x --> f x"
```
```   165     by (rule DERIV_isCont [unfolded isCont_def])
```
```   166
```
```   167   from neq have "0 < norm (f x)" by simp
```
```   168   with LIM_D [OF lim_f] obtain s
```
```   169     where s: "0 < s"
```
```   170     and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
```
```   171                   \<Longrightarrow> norm (f z - f x) < norm (f x)"
```
```   172     by fast
```
```   173
```
```   174   show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
```
```   175   proof (rule LIM_equal2 [OF s])
```
```   176     fix z
```
```   177     assume "z \<noteq> x" "norm (z - x) < s"
```
```   178     hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
```
```   179     hence "f z \<noteq> 0" by auto
```
```   180     thus "(inverse (f z) - inverse (f x)) / (z - x) =
```
```   181           - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"
```
```   182       using neq by (rule DERIV_inverse_lemma)
```
```   183   next
```
```   184     from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"
```
```   185       by (unfold DERIV_iff2)
```
```   186     thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
```
```   187           -- x --> ?E"
```
```   188       by (intro LIM_mult LIM_inverse LIM_minus LIM_const lim_f neq)
```
```   189   qed
```
```   190 qed
```
```   191
```
```   192 lemma DERIV_divide:
```
```   193   "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
```
```   194    \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
```
```   195 apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
```
```   196           D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
```
```   197 apply (erule subst)
```
```   198 apply (unfold divide_inverse)
```
```   199 apply (erule DERIV_mult')
```
```   200 apply (erule (1) DERIV_inverse')
```
```   201 apply (simp add: left_diff_distrib nonzero_inverse_mult_distrib)
```
```   202 apply (simp add: mult_ac)
```
```   203 done
```
```   204
```
```   205 lemma DERIV_power_Suc:
```
```   206   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
```
```   207   assumes f: "DERIV f x :> D"
```
```   208   shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (of_nat n + 1) * (D * f x ^ n)"
```
```   209 proof (induct n)
```
```   210 case 0
```
```   211   show ?case by (simp add: power_Suc f)
```
```   212 case (Suc k)
```
```   213   from DERIV_mult' [OF f Suc] show ?case
```
```   214     apply (simp only: of_nat_Suc left_distrib mult_1_left)
```
```   215     apply (simp only: power_Suc right_distrib mult_ac)
```
```   216     done
```
```   217 qed
```
```   218
```
```   219 lemma DERIV_power:
```
```   220   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
```
```   221   assumes f: "DERIV f x :> D"
```
```   222   shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
```
```   223 by (cases "n", simp, simp add: DERIV_power_Suc f)
```
```   224
```
```   225
```
```   226 (* ------------------------------------------------------------------------ *)
```
```   227 (* Caratheodory formulation of derivative at a point: standard proof        *)
```
```   228 (* ------------------------------------------------------------------------ *)
```
```   229
```
```   230 lemma nonzero_mult_divide_cancel_right:
```
```   231   "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
```
```   232 proof -
```
```   233   assume b: "b \<noteq> 0"
```
```   234   have "a * b / b = a * (b / b)" by simp
```
```   235   also have "\<dots> = a" by (simp add: divide_self b)
```
```   236   finally show "a * b / b = a" .
```
```   237 qed
```
```   238
```
```   239 lemma CARAT_DERIV:
```
```   240      "(DERIV f x :> l) =
```
```   241       (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
```
```   242       (is "?lhs = ?rhs")
```
```   243 proof
```
```   244   assume der: "DERIV f x :> l"
```
```   245   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
```
```   246   proof (intro exI conjI)
```
```   247     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
```
```   248     show "\<forall>z. f z - f x = ?g z * (z-x)"
```
```   249       by (simp add: nonzero_mult_divide_cancel_right)
```
```   250     show "isCont ?g x" using der
```
```   251       by (simp add: isCont_iff DERIV_iff diff_minus
```
```   252                cong: LIM_equal [rule_format])
```
```   253     show "?g x = l" by simp
```
```   254   qed
```
```   255 next
```
```   256   assume "?rhs"
```
```   257   then obtain g where
```
```   258     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
```
```   259   thus "(DERIV f x :> l)"
```
```   260      by (auto simp add: isCont_iff DERIV_iff nonzero_mult_divide_cancel_right
```
```   261               cong: LIM_cong)
```
```   262 qed
```
```   263
```
```   264 lemma DERIV_chain':
```
```   265   assumes f: "DERIV f x :> D"
```
```   266   assumes g: "DERIV g (f x) :> E"
```
```   267   shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
```
```   268 proof (unfold DERIV_iff2)
```
```   269   obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
```
```   270     and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
```
```   271     using CARAT_DERIV [THEN iffD1, OF g] by fast
```
```   272   from f have "f -- x --> f x"
```
```   273     by (rule DERIV_isCont [unfolded isCont_def])
```
```   274   with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
```
```   275     by (rule isCont_LIM_compose)
```
```   276   hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
```
```   277           -- x --> d (f x) * D"
```
```   278     by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
```
```   279   thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
```
```   280     by (simp add: d dfx real_scaleR_def)
```
```   281 qed
```
```   282
```
```   283 (* let's do the standard proof though theorem *)
```
```   284 (* LIM_mult2 follows from a NS proof          *)
```
```   285
```
```   286 lemma DERIV_cmult:
```
```   287       "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
```
```   288 by (drule DERIV_mult' [OF DERIV_const], simp)
```
```   289
```
```   290 (* standard version *)
```
```   291 lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
```
```   292 by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
```
```   293
```
```   294 lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
```
```   295 by (auto dest: DERIV_chain simp add: o_def)
```
```   296
```
```   297 (*derivative of linear multiplication*)
```
```   298 lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
```
```   299 by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
```
```   300
```
```   301 lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
```
```   302 apply (cut_tac DERIV_power [OF DERIV_ident])
```
```   303 apply (simp add: real_scaleR_def real_of_nat_def)
```
```   304 done
```
```   305
```
```   306 text{*Power of -1*}
```
```   307
```
```   308 lemma DERIV_inverse:
```
```   309   fixes x :: "'a::{real_normed_field,recpower}"
```
```   310   shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
```
```   311 by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc)
```
```   312
```
```   313 text{*Derivative of inverse*}
```
```   314 lemma DERIV_inverse_fun:
```
```   315   fixes x :: "'a::{real_normed_field,recpower}"
```
```   316   shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
```
```   317       ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
```
```   318 by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib)
```
```   319
```
```   320 text{*Derivative of quotient*}
```
```   321 lemma DERIV_quotient:
```
```   322   fixes x :: "'a::{real_normed_field,recpower}"
```
```   323   shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
```
```   324        ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
```
```   325 by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)
```
```   326
```
```   327
```
```   328 subsection {* Differentiability predicate *}
```
```   329
```
```   330 lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
```
```   331 by (simp add: differentiable_def)
```
```   332
```
```   333 lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
```
```   334 by (force simp add: differentiable_def)
```
```   335
```
```   336 lemma differentiable_const: "(\<lambda>z. a) differentiable x"
```
```   337   apply (unfold differentiable_def)
```
```   338   apply (rule_tac x=0 in exI)
```
```   339   apply simp
```
```   340   done
```
```   341
```
```   342 lemma differentiable_sum:
```
```   343   assumes "f differentiable x"
```
```   344   and "g differentiable x"
```
```   345   shows "(\<lambda>x. f x + g x) differentiable x"
```
```   346 proof -
```
```   347   from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
```
```   348   then obtain df where "DERIV f x :> df" ..
```
```   349   moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
```
```   350   then obtain dg where "DERIV g x :> dg" ..
```
```   351   ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
```
```   352   hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
```
```   353   thus ?thesis by (fold differentiable_def)
```
```   354 qed
```
```   355
```
```   356 lemma differentiable_diff:
```
```   357   assumes "f differentiable x"
```
```   358   and "g differentiable x"
```
```   359   shows "(\<lambda>x. f x - g x) differentiable x"
```
```   360 proof -
```
```   361   from prems have "f differentiable x" by simp
```
```   362   moreover
```
```   363   from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
```
```   364   then obtain dg where "DERIV g x :> dg" ..
```
```   365   then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
```
```   366   hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
```
```   367   hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
```
```   368   ultimately
```
```   369   show ?thesis
```
```   370     by (auto simp: diff_def dest: differentiable_sum)
```
```   371 qed
```
```   372
```
```   373 lemma differentiable_mult:
```
```   374   assumes "f differentiable x"
```
```   375   and "g differentiable x"
```
```   376   shows "(\<lambda>x. f x * g x) differentiable x"
```
```   377 proof -
```
```   378   from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
```
```   379   then obtain df where "DERIV f x :> df" ..
```
```   380   moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
```
```   381   then obtain dg where "DERIV g x :> dg" ..
```
```   382   ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
```
```   383   hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
```
```   384   thus ?thesis by (fold differentiable_def)
```
```   385 qed
```
```   386
```
```   387
```
```   388 subsection {* Nested Intervals and Bisection *}
```
```   389
```
```   390 text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
```
```   391      All considerably tidied by lcp.*}
```
```   392
```
```   393 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)"
```
```   394 apply (induct "no")
```
```   395 apply (auto intro: order_trans)
```
```   396 done
```
```   397
```
```   398 lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   399          \<forall>n. g(Suc n) \<le> g(n);
```
```   400          \<forall>n. f(n) \<le> g(n) |]
```
```   401       ==> Bseq (f :: nat \<Rightarrow> real)"
```
```   402 apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
```
```   403 apply (induct_tac "n")
```
```   404 apply (auto intro: order_trans)
```
```   405 apply (rule_tac y = "g (Suc na)" in order_trans)
```
```   406 apply (induct_tac [2] "na")
```
```   407 apply (auto intro: order_trans)
```
```   408 done
```
```   409
```
```   410 lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   411          \<forall>n. g(Suc n) \<le> g(n);
```
```   412          \<forall>n. f(n) \<le> g(n) |]
```
```   413       ==> Bseq (g :: nat \<Rightarrow> real)"
```
```   414 apply (subst Bseq_minus_iff [symmetric])
```
```   415 apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
```
```   416 apply auto
```
```   417 done
```
```   418
```
```   419 lemma f_inc_imp_le_lim:
```
```   420   fixes f :: "nat \<Rightarrow> real"
```
```   421   shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
```
```   422 apply (rule linorder_not_less [THEN iffD1])
```
```   423 apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
```
```   424 apply (drule real_less_sum_gt_zero)
```
```   425 apply (drule_tac x = "f n + - lim f" in spec, safe)
```
```   426 apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
```
```   427 apply (subgoal_tac "lim f \<le> f (no + n) ")
```
```   428 apply (drule_tac no=no and m=n in lemma_f_mono_add)
```
```   429 apply (auto simp add: add_commute)
```
```   430 apply (induct_tac "no")
```
```   431 apply simp
```
```   432 apply (auto intro: order_trans simp add: diff_minus abs_if)
```
```   433 done
```
```   434
```
```   435 lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
```
```   436 apply (rule LIMSEQ_minus [THEN limI])
```
```   437 apply (simp add: convergent_LIMSEQ_iff)
```
```   438 done
```
```   439
```
```   440 lemma g_dec_imp_lim_le:
```
```   441   fixes g :: "nat \<Rightarrow> real"
```
```   442   shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
```
```   443 apply (subgoal_tac "- (g n) \<le> - (lim g) ")
```
```   444 apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
```
```   445 apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
```
```   446 done
```
```   447
```
```   448 lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   449          \<forall>n. g(Suc n) \<le> g(n);
```
```   450          \<forall>n. f(n) \<le> g(n) |]
```
```   451       ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
```
```   452                             ((\<forall>n. m \<le> g(n)) & g ----> m)"
```
```   453 apply (subgoal_tac "monoseq f & monoseq g")
```
```   454 prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
```
```   455 apply (subgoal_tac "Bseq f & Bseq g")
```
```   456 prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
```
```   457 apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
```
```   458 apply (rule_tac x = "lim f" in exI)
```
```   459 apply (rule_tac x = "lim g" in exI)
```
```   460 apply (auto intro: LIMSEQ_le)
```
```   461 apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
```
```   462 done
```
```   463
```
```   464 lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   465          \<forall>n. g(Suc n) \<le> g(n);
```
```   466          \<forall>n. f(n) \<le> g(n);
```
```   467          (%n. f(n) - g(n)) ----> 0 |]
```
```   468       ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
```
```   469                 ((\<forall>n. l \<le> g(n)) & g ----> l)"
```
```   470 apply (drule lemma_nest, auto)
```
```   471 apply (subgoal_tac "l = m")
```
```   472 apply (drule_tac [2] X = f in LIMSEQ_diff)
```
```   473 apply (auto intro: LIMSEQ_unique)
```
```   474 done
```
```   475
```
```   476 text{*The universal quantifiers below are required for the declaration
```
```   477   of @{text Bolzano_nest_unique} below.*}
```
```   478
```
```   479 lemma Bolzano_bisect_le:
```
```   480  "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
```
```   481 apply (rule allI)
```
```   482 apply (induct_tac "n")
```
```   483 apply (auto simp add: Let_def split_def)
```
```   484 done
```
```   485
```
```   486 lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
```
```   487    \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
```
```   488 apply (rule allI)
```
```   489 apply (induct_tac "n")
```
```   490 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
```
```   491 done
```
```   492
```
```   493 lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
```
```   494    \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
```
```   495 apply (rule allI)
```
```   496 apply (induct_tac "n")
```
```   497 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
```
```   498 done
```
```   499
```
```   500 lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
```
```   501 apply (auto)
```
```   502 apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
```
```   503 apply (simp)
```
```   504 done
```
```   505
```
```   506 lemma Bolzano_bisect_diff:
```
```   507      "a \<le> b ==>
```
```   508       snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
```
```   509       (b-a) / (2 ^ n)"
```
```   510 apply (induct "n")
```
```   511 apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
```
```   512 done
```
```   513
```
```   514 lemmas Bolzano_nest_unique =
```
```   515     lemma_nest_unique
```
```   516     [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
```
```   517
```
```   518
```
```   519 lemma not_P_Bolzano_bisect:
```
```   520   assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
```
```   521       and notP: "~ P(a,b)"
```
```   522       and le:   "a \<le> b"
```
```   523   shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
```
```   524 proof (induct n)
```
```   525   case 0 thus ?case by simp
```
```   526  next
```
```   527   case (Suc n)
```
```   528   thus ?case
```
```   529  by (auto simp del: surjective_pairing [symmetric]
```
```   530              simp add: Let_def split_def Bolzano_bisect_le [OF le]
```
```   531      P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
```
```   532 qed
```
```   533
```
```   534 (*Now we re-package P_prem as a formula*)
```
```   535 lemma not_P_Bolzano_bisect':
```
```   536      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
```
```   537          ~ P(a,b);  a \<le> b |] ==>
```
```   538       \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
```
```   539 by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
```
```   540
```
```   541
```
```   542
```
```   543 lemma lemma_BOLZANO:
```
```   544      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
```
```   545          \<forall>x. \<exists>d::real. 0 < d &
```
```   546                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
```
```   547          a \<le> b |]
```
```   548       ==> P(a,b)"
```
```   549 apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
```
```   550 apply (rule LIMSEQ_minus_cancel)
```
```   551 apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
```
```   552 apply (rule ccontr)
```
```   553 apply (drule not_P_Bolzano_bisect', assumption+)
```
```   554 apply (rename_tac "l")
```
```   555 apply (drule_tac x = l in spec, clarify)
```
```   556 apply (simp add: LIMSEQ_def)
```
```   557 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
```
```   558 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
```
```   559 apply (drule real_less_half_sum, auto)
```
```   560 apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
```
```   561 apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
```
```   562 apply safe
```
```   563 apply (simp_all (no_asm_simp))
```
```   564 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)
```
```   565 apply (simp (no_asm_simp) add: abs_if)
```
```   566 apply (rule real_sum_of_halves [THEN subst])
```
```   567 apply (rule add_strict_mono)
```
```   568 apply (simp_all add: diff_minus [symmetric])
```
```   569 done
```
```   570
```
```   571
```
```   572 lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
```
```   573        (\<forall>x. \<exists>d::real. 0 < d &
```
```   574                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
```
```   575       --> (\<forall>a b. a \<le> b --> P(a,b))"
```
```   576 apply clarify
```
```   577 apply (blast intro: lemma_BOLZANO)
```
```   578 done
```
```   579
```
```   580
```
```   581 subsection {* Intermediate Value Theorem *}
```
```   582
```
```   583 text {*Prove Contrapositive by Bisection*}
```
```   584
```
```   585 lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
```
```   586          a \<le> b;
```
```   587          (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
```
```   588       ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
```
```   589 apply (rule contrapos_pp, assumption)
```
```   590 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)
```
```   591 apply safe
```
```   592 apply simp_all
```
```   593 apply (simp add: isCont_iff LIM_def)
```
```   594 apply (rule ccontr)
```
```   595 apply (subgoal_tac "a \<le> x & x \<le> b")
```
```   596  prefer 2
```
```   597  apply simp
```
```   598  apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
```
```   599 apply (drule_tac x = x in spec)+
```
```   600 apply simp
```
```   601 apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
```
```   602 apply safe
```
```   603 apply simp
```
```   604 apply (drule_tac x = s in spec, clarify)
```
```   605 apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
```
```   606 apply (drule_tac x = "ba-x" in spec)
```
```   607 apply (simp_all add: abs_if)
```
```   608 apply (drule_tac x = "aa-x" in spec)
```
```   609 apply (case_tac "x \<le> aa", simp_all)
```
```   610 done
```
```   611
```
```   612 lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
```
```   613          a \<le> b;
```
```   614          (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
```
```   615       |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
```
```   616 apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
```
```   617 apply (drule IVT [where f = "%x. - f x"], assumption)
```
```   618 apply (auto intro: isCont_minus)
```
```   619 done
```
```   620
```
```   621 (*HOL style here: object-level formulations*)
```
```   622 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
```
```   623       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```   624       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```   625 apply (blast intro: IVT)
```
```   626 done
```
```   627
```
```   628 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
```
```   629       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```   630       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```   631 apply (blast intro: IVT2)
```
```   632 done
```
```   633
```
```   634 text{*By bisection, function continuous on closed interval is bounded above*}
```
```   635
```
```   636 lemma isCont_bounded:
```
```   637      "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   638       ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
```
```   639 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)
```
```   640 apply safe
```
```   641 apply simp_all
```
```   642 apply (rename_tac x xa ya M Ma)
```
```   643 apply (cut_tac x = M and y = Ma in linorder_linear, safe)
```
```   644 apply (rule_tac x = Ma in exI, clarify)
```
```   645 apply (cut_tac x = xb and y = xa in linorder_linear, force)
```
```   646 apply (rule_tac x = M in exI, clarify)
```
```   647 apply (cut_tac x = xb and y = xa in linorder_linear, force)
```
```   648 apply (case_tac "a \<le> x & x \<le> b")
```
```   649 apply (rule_tac [2] x = 1 in exI)
```
```   650 prefer 2 apply force
```
```   651 apply (simp add: LIM_def isCont_iff)
```
```   652 apply (drule_tac x = x in spec, auto)
```
```   653 apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
```
```   654 apply (drule_tac x = 1 in spec, auto)
```
```   655 apply (rule_tac x = s in exI, clarify)
```
```   656 apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
```
```   657 apply (drule_tac x = "xa-x" in spec)
```
```   658 apply (auto simp add: abs_ge_self)
```
```   659 done
```
```   660
```
```   661 text{*Refine the above to existence of least upper bound*}
```
```   662
```
```   663 lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
```
```   664       (\<exists>t. isLub UNIV S t)"
```
```   665 by (blast intro: reals_complete)
```
```   666
```
```   667 lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   668          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
```
```   669                    (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
```
```   670 apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
```
```   671         in lemma_reals_complete)
```
```   672 apply auto
```
```   673 apply (drule isCont_bounded, assumption)
```
```   674 apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
```
```   675 apply (rule exI, auto)
```
```   676 apply (auto dest!: spec simp add: linorder_not_less)
```
```   677 done
```
```   678
```
```   679 text{*Now show that it attains its upper bound*}
```
```   680
```
```   681 lemma isCont_eq_Ub:
```
```   682   assumes le: "a \<le> b"
```
```   683       and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
```
```   684   shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
```
```   685              (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
```
```   686 proof -
```
```   687   from isCont_has_Ub [OF le con]
```
```   688   obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
```
```   689              and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
```
```   690   show ?thesis
```
```   691   proof (intro exI, intro conjI)
```
```   692     show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
```
```   693     show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
```
```   694     proof (rule ccontr)
```
```   695       assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
```
```   696       with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
```
```   697         by (fastsimp simp add: linorder_not_le [symmetric])
```
```   698       hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
```
```   699         by (auto simp add: isCont_inverse isCont_diff con)
```
```   700       from isCont_bounded [OF le this]
```
```   701       obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
```
```   702       have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
```
```   703         by (simp add: M3 compare_rls)
```
```   704       have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
```
```   705         by (auto intro: order_le_less_trans [of _ k])
```
```   706       with Minv
```
```   707       have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
```
```   708         by (intro strip less_imp_inverse_less, simp_all)
```
```   709       hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
```
```   710         by simp
```
```   711       have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
```
```   712         by (simp, arith)
```
```   713       from M2 [OF this]
```
```   714       obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
```
```   715       thus False using invlt [of x] by force
```
```   716     qed
```
```   717   qed
```
```   718 qed
```
```   719
```
```   720
```
```   721 text{*Same theorem for lower bound*}
```
```   722
```
```   723 lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   724          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
```
```   725                    (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
```
```   726 apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
```
```   727 prefer 2 apply (blast intro: isCont_minus)
```
```   728 apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
```
```   729 apply safe
```
```   730 apply auto
```
```   731 done
```
```   732
```
```   733
```
```   734 text{*Another version.*}
```
```   735
```
```   736 lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   737       ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
```
```   738           (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
```
```   739 apply (frule isCont_eq_Lb)
```
```   740 apply (frule_tac [2] isCont_eq_Ub)
```
```   741 apply (assumption+, safe)
```
```   742 apply (rule_tac x = "f x" in exI)
```
```   743 apply (rule_tac x = "f xa" in exI, simp, safe)
```
```   744 apply (cut_tac x = x and y = xa in linorder_linear, safe)
```
```   745 apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
```
```   746 apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
```
```   747 apply (rule_tac [2] x = xb in exI)
```
```   748 apply (rule_tac [4] x = xb in exI, simp_all)
```
```   749 done
```
```   750
```
```   751
```
```   752 text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
```
```   753
```
```   754 lemma DERIV_left_inc:
```
```   755   fixes f :: "real => real"
```
```   756   assumes der: "DERIV f x :> l"
```
```   757       and l:   "0 < l"
```
```   758   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
```
```   759 proof -
```
```   760   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
```
```   761   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)"
```
```   762     by (simp add: diff_minus)
```
```   763   then obtain s
```
```   764         where s:   "0 < s"
```
```   765           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
```
```   766     by auto
```
```   767   thus ?thesis
```
```   768   proof (intro exI conjI strip)
```
```   769     show "0<s" .
```
```   770     fix h::real
```
```   771     assume "0 < h" "h < s"
```
```   772     with all [of h] show "f x < f (x+h)"
```
```   773     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   774     split add: split_if_asm)
```
```   775       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
```
```   776       with l
```
```   777       have "0 < (f (x+h) - f x) / h" by arith
```
```   778       thus "f x < f (x+h)"
```
```   779   by (simp add: pos_less_divide_eq h)
```
```   780     qed
```
```   781   qed
```
```   782 qed
```
```   783
```
```   784 lemma DERIV_left_dec:
```
```   785   fixes f :: "real => real"
```
```   786   assumes der: "DERIV f x :> l"
```
```   787       and l:   "l < 0"
```
```   788   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
```
```   789 proof -
```
```   790   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
```
```   791   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)"
```
```   792     by (simp add: diff_minus)
```
```   793   then obtain s
```
```   794         where s:   "0 < s"
```
```   795           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
```
```   796     by auto
```
```   797   thus ?thesis
```
```   798   proof (intro exI conjI strip)
```
```   799     show "0<s" .
```
```   800     fix h::real
```
```   801     assume "0 < h" "h < s"
```
```   802     with all [of "-h"] show "f x < f (x-h)"
```
```   803     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   804     split add: split_if_asm)
```
```   805       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
```
```   806       with l
```
```   807       have "0 < (f (x-h) - f x) / h" by arith
```
```   808       thus "f x < f (x-h)"
```
```   809   by (simp add: pos_less_divide_eq h)
```
```   810     qed
```
```   811   qed
```
```   812 qed
```
```   813
```
```   814 lemma DERIV_local_max:
```
```   815   fixes f :: "real => real"
```
```   816   assumes der: "DERIV f x :> l"
```
```   817       and d:   "0 < d"
```
```   818       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
```
```   819   shows "l = 0"
```
```   820 proof (cases rule: linorder_cases [of l 0])
```
```   821   case equal show ?thesis .
```
```   822 next
```
```   823   case less
```
```   824   from DERIV_left_dec [OF der less]
```
```   825   obtain d' where d': "0 < d'"
```
```   826              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
```
```   827   from real_lbound_gt_zero [OF d d']
```
```   828   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   829   with lt le [THEN spec [where x="x-e"]]
```
```   830   show ?thesis by (auto simp add: abs_if)
```
```   831 next
```
```   832   case greater
```
```   833   from DERIV_left_inc [OF der greater]
```
```   834   obtain d' where d': "0 < d'"
```
```   835              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
```
```   836   from real_lbound_gt_zero [OF d d']
```
```   837   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   838   with lt le [THEN spec [where x="x+e"]]
```
```   839   show ?thesis by (auto simp add: abs_if)
```
```   840 qed
```
```   841
```
```   842
```
```   843 text{*Similar theorem for a local minimum*}
```
```   844 lemma DERIV_local_min:
```
```   845   fixes f :: "real => real"
```
```   846   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
```
```   847 by (drule DERIV_minus [THEN DERIV_local_max], auto)
```
```   848
```
```   849
```
```   850 text{*In particular, if a function is locally flat*}
```
```   851 lemma DERIV_local_const:
```
```   852   fixes f :: "real => real"
```
```   853   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
```
```   854 by (auto dest!: DERIV_local_max)
```
```   855
```
```   856 text{*Lemma about introducing open ball in open interval*}
```
```   857 lemma lemma_interval_lt:
```
```   858      "[| a < x;  x < b |]
```
```   859       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
```
```   860 apply (simp add: abs_less_iff)
```
```   861 apply (insert linorder_linear [of "x-a" "b-x"], safe)
```
```   862 apply (rule_tac x = "x-a" in exI)
```
```   863 apply (rule_tac [2] x = "b-x" in exI, auto)
```
```   864 done
```
```   865
```
```   866 lemma lemma_interval: "[| a < x;  x < b |] ==>
```
```   867         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
```
```   868 apply (drule lemma_interval_lt, auto)
```
```   869 apply (auto intro!: exI)
```
```   870 done
```
```   871
```
```   872 text{*Rolle's Theorem.
```
```   873    If @{term f} is defined and continuous on the closed interval
```
```   874    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
```
```   875    and @{term "f(a) = f(b)"},
```
```   876    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
```
```   877 theorem Rolle:
```
```   878   assumes lt: "a < b"
```
```   879       and eq: "f(a) = f(b)"
```
```   880       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```   881       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```   882   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
```
```   883 proof -
```
```   884   have le: "a \<le> b" using lt by simp
```
```   885   from isCont_eq_Ub [OF le con]
```
```   886   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
```
```   887              and alex: "a \<le> x" and xleb: "x \<le> b"
```
```   888     by blast
```
```   889   from isCont_eq_Lb [OF le con]
```
```   890   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
```
```   891               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
```
```   892     by blast
```
```   893   show ?thesis
```
```   894   proof cases
```
```   895     assume axb: "a < x & x < b"
```
```   896         --{*@{term f} attains its maximum within the interval*}
```
```   897     hence ax: "a<x" and xb: "x<b" by auto
```
```   898     from lemma_interval [OF ax xb]
```
```   899     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   900       by blast
```
```   901     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
```
```   902       by blast
```
```   903     from differentiableD [OF dif [OF axb]]
```
```   904     obtain l where der: "DERIV f x :> l" ..
```
```   905     have "l=0" by (rule DERIV_local_max [OF der d bound'])
```
```   906         --{*the derivative at a local maximum is zero*}
```
```   907     thus ?thesis using ax xb der by auto
```
```   908   next
```
```   909     assume notaxb: "~ (a < x & x < b)"
```
```   910     hence xeqab: "x=a | x=b" using alex xleb by arith
```
```   911     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
```
```   912     show ?thesis
```
```   913     proof cases
```
```   914       assume ax'b: "a < x' & x' < b"
```
```   915         --{*@{term f} attains its minimum within the interval*}
```
```   916       hence ax': "a<x'" and x'b: "x'<b" by auto
```
```   917       from lemma_interval [OF ax' x'b]
```
```   918       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   919   by blast
```
```   920       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
```
```   921   by blast
```
```   922       from differentiableD [OF dif [OF ax'b]]
```
```   923       obtain l where der: "DERIV f x' :> l" ..
```
```   924       have "l=0" by (rule DERIV_local_min [OF der d bound'])
```
```   925         --{*the derivative at a local minimum is zero*}
```
```   926       thus ?thesis using ax' x'b der by auto
```
```   927     next
```
```   928       assume notax'b: "~ (a < x' & x' < b)"
```
```   929         --{*@{term f} is constant througout the interval*}
```
```   930       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
```
```   931       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
```
```   932       from dense [OF lt]
```
```   933       obtain r where ar: "a < r" and rb: "r < b" by blast
```
```   934       from lemma_interval [OF ar rb]
```
```   935       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   936   by blast
```
```   937       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
```
```   938       proof (clarify)
```
```   939         fix z::real
```
```   940         assume az: "a \<le> z" and zb: "z \<le> b"
```
```   941         show "f z = f b"
```
```   942         proof (rule order_antisym)
```
```   943           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
```
```   944           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
```
```   945         qed
```
```   946       qed
```
```   947       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
```
```   948       proof (intro strip)
```
```   949         fix y::real
```
```   950         assume lt: "\<bar>r-y\<bar> < d"
```
```   951         hence "f y = f b" by (simp add: eq_fb bound)
```
```   952         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
```
```   953       qed
```
```   954       from differentiableD [OF dif [OF conjI [OF ar rb]]]
```
```   955       obtain l where der: "DERIV f r :> l" ..
```
```   956       have "l=0" by (rule DERIV_local_const [OF der d bound'])
```
```   957         --{*the derivative of a constant function is zero*}
```
```   958       thus ?thesis using ar rb der by auto
```
```   959     qed
```
```   960   qed
```
```   961 qed
```
```   962
```
```   963
```
```   964 subsection{*Mean Value Theorem*}
```
```   965
```
```   966 lemma lemma_MVT:
```
```   967      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
```
```   968 proof cases
```
```   969   assume "a=b" thus ?thesis by simp
```
```   970 next
```
```   971   assume "a\<noteq>b"
```
```   972   hence ba: "b-a \<noteq> 0" by arith
```
```   973   show ?thesis
```
```   974     by (rule real_mult_left_cancel [OF ba, THEN iffD1],
```
```   975         simp add: right_diff_distrib,
```
```   976         simp add: left_diff_distrib)
```
```   977 qed
```
```   978
```
```   979 theorem MVT:
```
```   980   assumes lt:  "a < b"
```
```   981       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```   982       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```   983   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
```
```   984                    (f(b) - f(a) = (b-a) * l)"
```
```   985 proof -
```
```   986   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
```
```   987   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
```
```   988     by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
```
```   989   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
```
```   990   proof (clarify)
```
```   991     fix x::real
```
```   992     assume ax: "a < x" and xb: "x < b"
```
```   993     from differentiableD [OF dif [OF conjI [OF ax xb]]]
```
```   994     obtain l where der: "DERIV f x :> l" ..
```
```   995     show "?F differentiable x"
```
```   996       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
```
```   997           blast intro: DERIV_diff DERIV_cmult_Id der)
```
```   998   qed
```
```   999   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
```
```  1000   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
```
```  1001     by blast
```
```  1002   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
```
```  1003     by (rule DERIV_cmult_Id)
```
```  1004   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
```
```  1005                    :> 0 + (f b - f a) / (b - a)"
```
```  1006     by (rule DERIV_add [OF der])
```
```  1007   show ?thesis
```
```  1008   proof (intro exI conjI)
```
```  1009     show "a < z" .
```
```  1010     show "z < b" .
```
```  1011     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
```
```  1012     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
```
```  1013   qed
```
```  1014 qed
```
```  1015
```
```  1016
```
```  1017 text{*A function is constant if its derivative is 0 over an interval.*}
```
```  1018
```
```  1019 lemma DERIV_isconst_end:
```
```  1020   fixes f :: "real => real"
```
```  1021   shows "[| a < b;
```
```  1022          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1023          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1024         ==> f b = f a"
```
```  1025 apply (drule MVT, assumption)
```
```  1026 apply (blast intro: differentiableI)
```
```  1027 apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
```
```  1028 done
```
```  1029
```
```  1030 lemma DERIV_isconst1:
```
```  1031   fixes f :: "real => real"
```
```  1032   shows "[| a < b;
```
```  1033          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1034          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1035         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
```
```  1036 apply safe
```
```  1037 apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
```
```  1038 apply (drule_tac b = x in DERIV_isconst_end, auto)
```
```  1039 done
```
```  1040
```
```  1041 lemma DERIV_isconst2:
```
```  1042   fixes f :: "real => real"
```
```  1043   shows "[| a < b;
```
```  1044          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1045          \<forall>x. a < x & x < b --> DERIV f x :> 0;
```
```  1046          a \<le> x; x \<le> b |]
```
```  1047         ==> f x = f a"
```
```  1048 apply (blast dest: DERIV_isconst1)
```
```  1049 done
```
```  1050
```
```  1051 lemma DERIV_isconst_all:
```
```  1052   fixes f :: "real => real"
```
```  1053   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
```
```  1054 apply (rule linorder_cases [of x y])
```
```  1055 apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
```
```  1056 done
```
```  1057
```
```  1058 lemma DERIV_const_ratio_const:
```
```  1059   fixes f :: "real => real"
```
```  1060   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
```
```  1061 apply (rule linorder_cases [of a b], auto)
```
```  1062 apply (drule_tac [!] f = f in MVT)
```
```  1063 apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
```
```  1064 apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
```
```  1065 done
```
```  1066
```
```  1067 lemma DERIV_const_ratio_const2:
```
```  1068   fixes f :: "real => real"
```
```  1069   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
```
```  1070 apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
```
```  1071 apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
```
```  1072 done
```
```  1073
```
```  1074 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
```
```  1075 by (simp)
```
```  1076
```
```  1077 lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
```
```  1078 by (simp)
```
```  1079
```
```  1080 text{*Gallileo's "trick": average velocity = av. of end velocities*}
```
```  1081
```
```  1082 lemma DERIV_const_average:
```
```  1083   fixes v :: "real => real"
```
```  1084   assumes neq: "a \<noteq> (b::real)"
```
```  1085       and der: "\<forall>x. DERIV v x :> k"
```
```  1086   shows "v ((a + b)/2) = (v a + v b)/2"
```
```  1087 proof (cases rule: linorder_cases [of a b])
```
```  1088   case equal with neq show ?thesis by simp
```
```  1089 next
```
```  1090   case less
```
```  1091   have "(v b - v a) / (b - a) = k"
```
```  1092     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1093   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1094   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
```
```  1095     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1096   ultimately show ?thesis using neq by force
```
```  1097 next
```
```  1098   case greater
```
```  1099   have "(v b - v a) / (b - a) = k"
```
```  1100     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1101   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1102   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
```
```  1103     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1104   ultimately show ?thesis using neq by (force simp add: add_commute)
```
```  1105 qed
```
```  1106
```
```  1107
```
```  1108 text{*Dull lemma: an continuous injection on an interval must have a
```
```  1109 strict maximum at an end point, not in the middle.*}
```
```  1110
```
```  1111 lemma lemma_isCont_inj:
```
```  1112   fixes f :: "real \<Rightarrow> real"
```
```  1113   assumes d: "0 < d"
```
```  1114       and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1115       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1116   shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
```
```  1117 proof (rule ccontr)
```
```  1118   assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
```
```  1119   hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
```
```  1120   show False
```
```  1121   proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
```
```  1122     case le
```
```  1123     from d cont all [of "x+d"]
```
```  1124     have flef: "f(x+d) \<le> f x"
```
```  1125      and xlex: "x - d \<le> x"
```
```  1126      and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
```
```  1127        by (auto simp add: abs_if)
```
```  1128     from IVT [OF le flef xlex cont']
```
```  1129     obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
```
```  1130     moreover
```
```  1131     hence "g(f x') = g (f(x+d))" by simp
```
```  1132     ultimately show False using d inj [of x'] inj [of "x+d"]
```
```  1133       by (simp add: abs_le_iff)
```
```  1134   next
```
```  1135     case ge
```
```  1136     from d cont all [of "x-d"]
```
```  1137     have flef: "f(x-d) \<le> f x"
```
```  1138      and xlex: "x \<le> x+d"
```
```  1139      and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
```
```  1140        by (auto simp add: abs_if)
```
```  1141     from IVT2 [OF ge flef xlex cont']
```
```  1142     obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
```
```  1143     moreover
```
```  1144     hence "g(f x') = g (f(x-d))" by simp
```
```  1145     ultimately show False using d inj [of x'] inj [of "x-d"]
```
```  1146       by (simp add: abs_le_iff)
```
```  1147   qed
```
```  1148 qed
```
```  1149
```
```  1150
```
```  1151 text{*Similar version for lower bound.*}
```
```  1152
```
```  1153 lemma lemma_isCont_inj2:
```
```  1154   fixes f g :: "real \<Rightarrow> real"
```
```  1155   shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
```
```  1156         \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
```
```  1157       ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
```
```  1158 apply (insert lemma_isCont_inj
```
```  1159           [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
```
```  1160 apply (simp add: isCont_minus linorder_not_le)
```
```  1161 done
```
```  1162
```
```  1163 text{*Show there's an interval surrounding @{term "f(x)"} in
```
```  1164 @{text "f[[x - d, x + d]]"} .*}
```
```  1165
```
```  1166 lemma isCont_inj_range:
```
```  1167   fixes f :: "real \<Rightarrow> real"
```
```  1168   assumes d: "0 < d"
```
```  1169       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1170       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1171   shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
```
```  1172 proof -
```
```  1173   have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
```
```  1174     by (auto simp add: abs_le_iff)
```
```  1175   from isCont_Lb_Ub [OF this]
```
```  1176   obtain L M
```
```  1177   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"
```
```  1178     and all2 [rule_format]:
```
```  1179            "\<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)"
```
```  1180     by auto
```
```  1181   with d have "L \<le> f x & f x \<le> M" by simp
```
```  1182   moreover have "L \<noteq> f x"
```
```  1183   proof -
```
```  1184     from lemma_isCont_inj2 [OF d inj cont]
```
```  1185     obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
```
```  1186     thus ?thesis using all1 [of u] by arith
```
```  1187   qed
```
```  1188   moreover have "f x \<noteq> M"
```
```  1189   proof -
```
```  1190     from lemma_isCont_inj [OF d inj cont]
```
```  1191     obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
```
```  1192     thus ?thesis using all1 [of u] by arith
```
```  1193   qed
```
```  1194   ultimately have "L < f x & f x < M" by arith
```
```  1195   hence "0 < f x - L" "0 < M - f x" by arith+
```
```  1196   from real_lbound_gt_zero [OF this]
```
```  1197   obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
```
```  1198   thus ?thesis
```
```  1199   proof (intro exI conjI)
```
```  1200     show "0<e" .
```
```  1201     show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
```
```  1202     proof (intro strip)
```
```  1203       fix y::real
```
```  1204       assume "\<bar>y - f x\<bar> \<le> e"
```
```  1205       with e have "L \<le> y \<and> y \<le> M" by arith
```
```  1206       from all2 [OF this]
```
```  1207       obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
```
```  1208       thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
```
```  1209         by (force simp add: abs_le_iff)
```
```  1210     qed
```
```  1211   qed
```
```  1212 qed
```
```  1213
```
```  1214
```
```  1215 text{*Continuity of inverse function*}
```
```  1216
```
```  1217 lemma isCont_inverse_function:
```
```  1218   fixes f g :: "real \<Rightarrow> real"
```
```  1219   assumes d: "0 < d"
```
```  1220       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1221       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1222   shows "isCont g (f x)"
```
```  1223 proof (simp add: isCont_iff LIM_eq)
```
```  1224   show "\<forall>r. 0 < r \<longrightarrow>
```
```  1225          (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
```
```  1226   proof (intro strip)
```
```  1227     fix r::real
```
```  1228     assume r: "0<r"
```
```  1229     from real_lbound_gt_zero [OF r d]
```
```  1230     obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
```
```  1231     with inj cont
```
```  1232     have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
```
```  1233                   "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
```
```  1234     from isCont_inj_range [OF e this]
```
```  1235     obtain e' where e': "0 < e'"
```
```  1236         and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
```
```  1237           by blast
```
```  1238     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"
```
```  1239     proof (intro exI conjI)
```
```  1240       show "0<e'" .
```
```  1241       show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
```
```  1242       proof (intro strip)
```
```  1243         fix z::real
```
```  1244         assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
```
```  1245         with e e_lt e_simps all [rule_format, of "f x + z"]
```
```  1246         show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
```
```  1247       qed
```
```  1248     qed
```
```  1249   qed
```
```  1250 qed
```
```  1251
```
```  1252 text {* Derivative of inverse function *}
```
```  1253
```
```  1254 lemma DERIV_inverse_function:
```
```  1255   fixes f g :: "real \<Rightarrow> real"
```
```  1256   assumes der: "DERIV f (g x) :> D"
```
```  1257   assumes neq: "D \<noteq> 0"
```
```  1258   assumes a: "a < x" and b: "x < b"
```
```  1259   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
```
```  1260   assumes cont: "isCont g x"
```
```  1261   shows "DERIV g x :> inverse D"
```
```  1262 unfolding DERIV_iff2
```
```  1263 proof (rule LIM_equal2)
```
```  1264   show "0 < min (x - a) (b - x)"
```
```  1265     using a b by simp
```
```  1266 next
```
```  1267   fix y
```
```  1268   assume "norm (y - x) < min (x - a) (b - x)"
```
```  1269   hence "a < y" and "y < b"
```
```  1270     by (simp_all add: abs_less_iff)
```
```  1271   thus "(g y - g x) / (y - x) =
```
```  1272         inverse ((f (g y) - x) / (g y - g x))"
```
```  1273     by (simp add: inj)
```
```  1274 next
```
```  1275   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
```
```  1276     by (rule der [unfolded DERIV_iff2])
```
```  1277   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
```
```  1278     using inj a b by simp
```
```  1279   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
```
```  1280   proof (safe intro!: exI)
```
```  1281     show "0 < min (x - a) (b - x)"
```
```  1282       using a b by simp
```
```  1283   next
```
```  1284     fix y
```
```  1285     assume "norm (y - x) < min (x - a) (b - x)"
```
```  1286     hence y: "a < y" "y < b"
```
```  1287       by (simp_all add: abs_less_iff)
```
```  1288     assume "g y = g x"
```
```  1289     hence "f (g y) = f (g x)" by simp
```
```  1290     hence "y = x" using inj y a b by simp
```
```  1291     also assume "y \<noteq> x"
```
```  1292     finally show False by simp
```
```  1293   qed
```
```  1294   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
```
```  1295     using cont 1 2 by (rule isCont_LIM_compose2)
```
```  1296   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
```
```  1297         -- x --> inverse D"
```
```  1298     using neq by (rule LIM_inverse)
```
```  1299 qed
```
```  1300
```
```  1301 theorem GMVT:
```
```  1302   fixes a b :: real
```
```  1303   assumes alb: "a < b"
```
```  1304   and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  1305   and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
```
```  1306   and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
```
```  1307   and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
```
```  1308   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)"
```
```  1309 proof -
```
```  1310   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
```
```  1311   from prems have "a < b" by simp
```
```  1312   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
```
```  1313   proof -
```
```  1314     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
```
```  1315     with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
```
```  1316       by (auto intro: isCont_mult)
```
```  1317     moreover
```
```  1318     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
```
```  1319     with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
```
```  1320       by (auto intro: isCont_mult)
```
```  1321     ultimately show ?thesis
```
```  1322       by (fastsimp intro: isCont_diff)
```
```  1323   qed
```
```  1324   moreover
```
```  1325   have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
```
```  1326   proof -
```
```  1327     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
```
```  1328     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)
```
```  1329     moreover
```
```  1330     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
```
```  1331     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)
```
```  1332     ultimately show ?thesis by (simp add: differentiable_diff)
```
```  1333   qed
```
```  1334   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)
```
```  1335   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" ..
```
```  1336   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
```
```  1337
```
```  1338   from cdef have cint: "a < c \<and> c < b" by auto
```
```  1339   with gd have "g differentiable c" by simp
```
```  1340   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
```
```  1341   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
```
```  1342
```
```  1343   from cdef have "a < c \<and> c < b" by auto
```
```  1344   with fd have "f differentiable c" by simp
```
```  1345   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
```
```  1346   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
```
```  1347
```
```  1348   from cdef have "DERIV ?h c :> l" by auto
```
```  1349   moreover
```
```  1350   {
```
```  1351     from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
```
```  1352       apply (insert DERIV_const [where k="f b - f a"])
```
```  1353       apply (drule meta_spec [of _ c])
```
```  1354       apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g])
```
```  1355       by simp_all
```
```  1356     moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
```
```  1357       apply (insert DERIV_const [where k="g b - g a"])
```
```  1358       apply (drule meta_spec [of _ c])
```
```  1359       apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f])
```
```  1360       by simp_all
```
```  1361     ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
```
```  1362       by (simp add: DERIV_diff)
```
```  1363   }
```
```  1364   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
```
```  1365
```
```  1366   {
```
```  1367     from cdef have "?h b - ?h a = (b - a) * l" by auto
```
```  1368     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1369     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1370   }
```
```  1371   moreover
```
```  1372   {
```
```  1373     have "?h b - ?h a =
```
```  1374          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
```
```  1375           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
```
```  1376       by (simp add: mult_ac add_ac right_diff_distrib)
```
```  1377     hence "?h b - ?h a = 0" by auto
```
```  1378   }
```
```  1379   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
```
```  1380   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
```
```  1381   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
```
```  1382   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
```
```  1383
```
```  1384   with g'cdef f'cdef cint show ?thesis by auto
```
```  1385 qed
```
```  1386
```
```  1387 lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
```
```  1388 by auto
```
```  1389
```
```  1390 end
```