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