src/HOL/Deriv.thy
 author huffman Tue Jan 13 06:57:08 2009 -0800 (2009-01-13) changeset 29470 1851088a1f87 parent 29169 6a5f1d8d7344 child 29472 a63a2e46cec9 permissions -rw-r--r--
convert Deriv.thy to use new Polynomial library (incomplete)
```     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 Polynomial
```
```    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 consts
```
```    24   Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
```
```    25 primrec
```
```    26   "Bolzano_bisect P a b 0 = (a,b)"
```
```    27   "Bolzano_bisect P a b (Suc n) =
```
```    28       (let (x,y) = Bolzano_bisect P a b n
```
```    29        in if P(x, (x+y)/2) then ((x+y)/2, y)
```
```    30                             else (x, (x+y)/2))"
```
```    31
```
```    32
```
```    33 subsection {* Derivatives *}
```
```    34
```
```    35 lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
```
```    36 by (simp add: deriv_def)
```
```    37
```
```    38 lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
```
```    39 by (simp add: deriv_def)
```
```    40
```
```    41 lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
```
```    42 by (simp add: deriv_def)
```
```    43
```
```    44 lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
```
```    45 by (simp add: deriv_def cong: LIM_cong)
```
```    46
```
```    47 lemma add_diff_add:
```
```    48   fixes a b c d :: "'a::ab_group_add"
```
```    49   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```    50 by simp
```
```    51
```
```    52 lemma DERIV_add:
```
```    53   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
```
```    54 by (simp only: deriv_def add_diff_add add_divide_distrib LIM_add)
```
```    55
```
```    56 lemma DERIV_minus:
```
```    57   "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
```
```    58 by (simp only: deriv_def minus_diff_minus divide_minus_left LIM_minus)
```
```    59
```
```    60 lemma DERIV_diff:
```
```    61   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
```
```    62 by (simp only: diff_def DERIV_add DERIV_minus)
```
```    63
```
```    64 lemma DERIV_add_minus:
```
```    65   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
```
```    66 by (simp only: DERIV_add DERIV_minus)
```
```    67
```
```    68 lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
```
```    69 proof (unfold isCont_iff)
```
```    70   assume "DERIV f x :> D"
```
```    71   hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"
```
```    72     by (rule DERIV_D)
```
```    73   hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
```
```    74     by (intro LIM_mult LIM_ident)
```
```    75   hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
```
```    76     by simp
```
```    77   hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
```
```    78     by (simp cong: LIM_cong)
```
```    79   thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
```
```    80     by (simp add: LIM_def)
```
```    81 qed
```
```    82
```
```    83 lemma DERIV_mult_lemma:
```
```    84   fixes a b c d :: "'a::real_field"
```
```    85   shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"
```
```    86 by (simp add: diff_minus add_divide_distrib [symmetric] ring_distribs)
```
```    87
```
```    88 lemma DERIV_mult':
```
```    89   assumes f: "DERIV f x :> D"
```
```    90   assumes g: "DERIV g x :> E"
```
```    91   shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
```
```    92 proof (unfold deriv_def)
```
```    93   from f have "isCont f x"
```
```    94     by (rule DERIV_isCont)
```
```    95   hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
```
```    96     by (simp only: isCont_iff)
```
```    97   hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +
```
```    98               ((f(x+h) - f x) / h) * g x)
```
```    99           -- 0 --> f x * E + D * g x"
```
```   100     by (intro LIM_add LIM_mult LIM_const DERIV_D f g)
```
```   101   thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)
```
```   102          -- 0 --> f x * E + D * g x"
```
```   103     by (simp only: DERIV_mult_lemma)
```
```   104 qed
```
```   105
```
```   106 lemma DERIV_mult:
```
```   107      "[| DERIV f x :> Da; DERIV g x :> Db |]
```
```   108       ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
```
```   109 by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
```
```   110
```
```   111 lemma DERIV_unique:
```
```   112       "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
```
```   113 apply (simp add: deriv_def)
```
```   114 apply (blast intro: LIM_unique)
```
```   115 done
```
```   116
```
```   117 text{*Differentiation of finite sum*}
```
```   118
```
```   119 lemma DERIV_sumr [rule_format (no_asm)]:
```
```   120      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
```
```   121       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
```
```   122 apply (induct "n")
```
```   123 apply (auto intro: DERIV_add)
```
```   124 done
```
```   125
```
```   126 text{*Alternative definition for differentiability*}
```
```   127
```
```   128 lemma DERIV_LIM_iff:
```
```   129      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
```
```   130       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
```
```   131 apply (rule iffI)
```
```   132 apply (drule_tac k="- a" in LIM_offset)
```
```   133 apply (simp add: diff_minus)
```
```   134 apply (drule_tac k="a" in LIM_offset)
```
```   135 apply (simp add: add_commute)
```
```   136 done
```
```   137
```
```   138 lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
```
```   139 by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
```
```   140
```
```   141 lemma inverse_diff_inverse:
```
```   142   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   143    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   144 by (simp add: ring_simps)
```
```   145
```
```   146 lemma DERIV_inverse_lemma:
```
```   147   "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
```
```   148    \<Longrightarrow> (inverse a - inverse b) / h
```
```   149      = - (inverse a * ((a - b) / h) * inverse b)"
```
```   150 by (simp add: inverse_diff_inverse)
```
```   151
```
```   152 lemma DERIV_inverse':
```
```   153   assumes der: "DERIV f x :> D"
```
```   154   assumes neq: "f x \<noteq> 0"
```
```   155   shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
```
```   156     (is "DERIV _ _ :> ?E")
```
```   157 proof (unfold DERIV_iff2)
```
```   158   from der have lim_f: "f -- x --> f x"
```
```   159     by (rule DERIV_isCont [unfolded isCont_def])
```
```   160
```
```   161   from neq have "0 < norm (f x)" by simp
```
```   162   with LIM_D [OF lim_f] obtain s
```
```   163     where s: "0 < s"
```
```   164     and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
```
```   165                   \<Longrightarrow> norm (f z - f x) < norm (f x)"
```
```   166     by fast
```
```   167
```
```   168   show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
```
```   169   proof (rule LIM_equal2 [OF s])
```
```   170     fix z
```
```   171     assume "z \<noteq> x" "norm (z - x) < s"
```
```   172     hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
```
```   173     hence "f z \<noteq> 0" by auto
```
```   174     thus "(inverse (f z) - inverse (f x)) / (z - x) =
```
```   175           - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"
```
```   176       using neq by (rule DERIV_inverse_lemma)
```
```   177   next
```
```   178     from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"
```
```   179       by (unfold DERIV_iff2)
```
```   180     thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
```
```   181           -- x --> ?E"
```
```   182       by (intro LIM_mult LIM_inverse LIM_minus LIM_const lim_f neq)
```
```   183   qed
```
```   184 qed
```
```   185
```
```   186 lemma DERIV_divide:
```
```   187   "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
```
```   188    \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
```
```   189 apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
```
```   190           D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
```
```   191 apply (erule subst)
```
```   192 apply (unfold divide_inverse)
```
```   193 apply (erule DERIV_mult')
```
```   194 apply (erule (1) DERIV_inverse')
```
```   195 apply (simp add: ring_distribs nonzero_inverse_mult_distrib)
```
```   196 apply (simp add: mult_ac)
```
```   197 done
```
```   198
```
```   199 lemma DERIV_power_Suc:
```
```   200   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
```
```   201   assumes f: "DERIV f x :> D"
```
```   202   shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"
```
```   203 proof (induct n)
```
```   204 case 0
```
```   205   show ?case by (simp add: power_Suc f)
```
```   206 case (Suc k)
```
```   207   from DERIV_mult' [OF f Suc] show ?case
```
```   208     apply (simp only: of_nat_Suc ring_distribs mult_1_left)
```
```   209     apply (simp only: power_Suc right_distrib mult_ac add_ac)
```
```   210     done
```
```   211 qed
```
```   212
```
```   213 lemma DERIV_power:
```
```   214   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
```
```   215   assumes f: "DERIV f x :> D"
```
```   216   shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
```
```   217 by (cases "n", simp, simp add: DERIV_power_Suc f)
```
```   218
```
```   219
```
```   220 (* ------------------------------------------------------------------------ *)
```
```   221 (* Caratheodory formulation of derivative at a point: standard proof        *)
```
```   222 (* ------------------------------------------------------------------------ *)
```
```   223
```
```   224 lemma CARAT_DERIV:
```
```   225      "(DERIV f x :> l) =
```
```   226       (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
```
```   227       (is "?lhs = ?rhs")
```
```   228 proof
```
```   229   assume der: "DERIV f x :> l"
```
```   230   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
```
```   231   proof (intro exI conjI)
```
```   232     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
```
```   233     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
```
```   234     show "isCont ?g x" using der
```
```   235       by (simp add: isCont_iff DERIV_iff diff_minus
```
```   236                cong: LIM_equal [rule_format])
```
```   237     show "?g x = l" by simp
```
```   238   qed
```
```   239 next
```
```   240   assume "?rhs"
```
```   241   then obtain g where
```
```   242     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
```
```   243   thus "(DERIV f x :> l)"
```
```   244      by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
```
```   245 qed
```
```   246
```
```   247 lemma DERIV_chain':
```
```   248   assumes f: "DERIV f x :> D"
```
```   249   assumes g: "DERIV g (f x) :> E"
```
```   250   shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
```
```   251 proof (unfold DERIV_iff2)
```
```   252   obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
```
```   253     and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
```
```   254     using CARAT_DERIV [THEN iffD1, OF g] by fast
```
```   255   from f have "f -- x --> f x"
```
```   256     by (rule DERIV_isCont [unfolded isCont_def])
```
```   257   with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
```
```   258     by (rule isCont_LIM_compose)
```
```   259   hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
```
```   260           -- x --> d (f x) * D"
```
```   261     by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
```
```   262   thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
```
```   263     by (simp add: d dfx real_scaleR_def)
```
```   264 qed
```
```   265
```
```   266 (* let's do the standard proof though theorem *)
```
```   267 (* LIM_mult2 follows from a NS proof          *)
```
```   268
```
```   269 lemma DERIV_cmult:
```
```   270       "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
```
```   271 by (drule DERIV_mult' [OF DERIV_const], simp)
```
```   272
```
```   273 (* standard version *)
```
```   274 lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
```
```   275 by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
```
```   276
```
```   277 lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
```
```   278 by (auto dest: DERIV_chain simp add: o_def)
```
```   279
```
```   280 (*derivative of linear multiplication*)
```
```   281 lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
```
```   282 by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
```
```   283
```
```   284 lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
```
```   285 apply (cut_tac DERIV_power [OF DERIV_ident])
```
```   286 apply (simp add: real_scaleR_def real_of_nat_def)
```
```   287 done
```
```   288
```
```   289 text{*Power of -1*}
```
```   290
```
```   291 lemma DERIV_inverse:
```
```   292   fixes x :: "'a::{real_normed_field,recpower}"
```
```   293   shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
```
```   294 by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc)
```
```   295
```
```   296 text{*Derivative of inverse*}
```
```   297 lemma DERIV_inverse_fun:
```
```   298   fixes x :: "'a::{real_normed_field,recpower}"
```
```   299   shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
```
```   300       ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
```
```   301 by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib)
```
```   302
```
```   303 text{*Derivative of quotient*}
```
```   304 lemma DERIV_quotient:
```
```   305   fixes x :: "'a::{real_normed_field,recpower}"
```
```   306   shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
```
```   307        ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
```
```   308 by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)
```
```   309
```
```   310
```
```   311 subsection {* Differentiability predicate *}
```
```   312
```
```   313 definition
```
```   314   differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
```
```   315     (infixl "differentiable" 60) where
```
```   316   "f differentiable x = (\<exists>D. DERIV f x :> D)"
```
```   317
```
```   318 lemma differentiableE [elim?]:
```
```   319   assumes "f differentiable x"
```
```   320   obtains df where "DERIV f x :> df"
```
```   321   using prems unfolding differentiable_def ..
```
```   322
```
```   323 lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
```
```   324 by (simp add: differentiable_def)
```
```   325
```
```   326 lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
```
```   327 by (force simp add: differentiable_def)
```
```   328
```
```   329 lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
```
```   330   by (rule DERIV_ident [THEN differentiableI])
```
```   331
```
```   332 lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
```
```   333   by (rule DERIV_const [THEN differentiableI])
```
```   334
```
```   335 lemma differentiable_compose:
```
```   336   assumes f: "f differentiable (g x)"
```
```   337   assumes g: "g differentiable x"
```
```   338   shows "(\<lambda>x. f (g x)) differentiable x"
```
```   339 proof -
```
```   340   from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
```
```   341   moreover
```
```   342   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   343   ultimately
```
```   344   have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
```
```   345   thus ?thesis by (rule differentiableI)
```
```   346 qed
```
```   347
```
```   348 lemma differentiable_sum [simp]:
```
```   349   assumes "f differentiable x"
```
```   350   and "g differentiable x"
```
```   351   shows "(\<lambda>x. f x + g x) differentiable x"
```
```   352 proof -
```
```   353   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   354   moreover
```
```   355   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   356   ultimately
```
```   357   have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
```
```   358   thus ?thesis by (rule differentiableI)
```
```   359 qed
```
```   360
```
```   361 lemma differentiable_minus [simp]:
```
```   362   assumes "f differentiable x"
```
```   363   shows "(\<lambda>x. - f x) differentiable x"
```
```   364 proof -
```
```   365   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   366   hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
```
```   367   thus ?thesis by (rule differentiableI)
```
```   368 qed
```
```   369
```
```   370 lemma differentiable_diff [simp]:
```
```   371   assumes "f differentiable x"
```
```   372   assumes "g differentiable x"
```
```   373   shows "(\<lambda>x. f x - g x) differentiable x"
```
```   374   unfolding diff_minus using prems by simp
```
```   375
```
```   376 lemma differentiable_mult [simp]:
```
```   377   assumes "f differentiable x"
```
```   378   assumes "g differentiable x"
```
```   379   shows "(\<lambda>x. f x * g x) differentiable x"
```
```   380 proof -
```
```   381   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   382   moreover
```
```   383   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
```
```   384   ultimately
```
```   385   have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
```
```   386   thus ?thesis by (rule differentiableI)
```
```   387 qed
```
```   388
```
```   389 lemma differentiable_inverse [simp]:
```
```   390   assumes "f differentiable x" and "f x \<noteq> 0"
```
```   391   shows "(\<lambda>x. inverse (f x)) differentiable x"
```
```   392 proof -
```
```   393   from `f differentiable x` obtain df where "DERIV f x :> df" ..
```
```   394   hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
```
```   395     using `f x \<noteq> 0` by (rule DERIV_inverse')
```
```   396   thus ?thesis by (rule differentiableI)
```
```   397 qed
```
```   398
```
```   399 lemma differentiable_divide [simp]:
```
```   400   assumes "f differentiable x"
```
```   401   assumes "g differentiable x" and "g x \<noteq> 0"
```
```   402   shows "(\<lambda>x. f x / g x) differentiable x"
```
```   403   unfolding divide_inverse using prems by simp
```
```   404
```
```   405 lemma differentiable_power [simp]:
```
```   406   fixes f :: "'a::{recpower,real_normed_field} \<Rightarrow> 'a"
```
```   407   assumes "f differentiable x"
```
```   408   shows "(\<lambda>x. f x ^ n) differentiable x"
```
```   409   by (induct n, simp, simp add: power_Suc prems)
```
```   410
```
```   411
```
```   412 subsection {* Nested Intervals and Bisection *}
```
```   413
```
```   414 text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
```
```   415      All considerably tidied by lcp.*}
```
```   416
```
```   417 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)"
```
```   418 apply (induct "no")
```
```   419 apply (auto intro: order_trans)
```
```   420 done
```
```   421
```
```   422 lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   423          \<forall>n. g(Suc n) \<le> g(n);
```
```   424          \<forall>n. f(n) \<le> g(n) |]
```
```   425       ==> Bseq (f :: nat \<Rightarrow> real)"
```
```   426 apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
```
```   427 apply (induct_tac "n")
```
```   428 apply (auto intro: order_trans)
```
```   429 apply (rule_tac y = "g (Suc na)" in order_trans)
```
```   430 apply (induct_tac [2] "na")
```
```   431 apply (auto intro: order_trans)
```
```   432 done
```
```   433
```
```   434 lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   435          \<forall>n. g(Suc n) \<le> g(n);
```
```   436          \<forall>n. f(n) \<le> g(n) |]
```
```   437       ==> Bseq (g :: nat \<Rightarrow> real)"
```
```   438 apply (subst Bseq_minus_iff [symmetric])
```
```   439 apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
```
```   440 apply auto
```
```   441 done
```
```   442
```
```   443 lemma f_inc_imp_le_lim:
```
```   444   fixes f :: "nat \<Rightarrow> real"
```
```   445   shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
```
```   446 apply (rule linorder_not_less [THEN iffD1])
```
```   447 apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
```
```   448 apply (drule real_less_sum_gt_zero)
```
```   449 apply (drule_tac x = "f n + - lim f" in spec, safe)
```
```   450 apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
```
```   451 apply (subgoal_tac "lim f \<le> f (no + n) ")
```
```   452 apply (drule_tac no=no and m=n in lemma_f_mono_add)
```
```   453 apply (auto simp add: add_commute)
```
```   454 apply (induct_tac "no")
```
```   455 apply simp
```
```   456 apply (auto intro: order_trans simp add: diff_minus abs_if)
```
```   457 done
```
```   458
```
```   459 lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
```
```   460 apply (rule LIMSEQ_minus [THEN limI])
```
```   461 apply (simp add: convergent_LIMSEQ_iff)
```
```   462 done
```
```   463
```
```   464 lemma g_dec_imp_lim_le:
```
```   465   fixes g :: "nat \<Rightarrow> real"
```
```   466   shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
```
```   467 apply (subgoal_tac "- (g n) \<le> - (lim g) ")
```
```   468 apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
```
```   469 apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
```
```   470 done
```
```   471
```
```   472 lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   473          \<forall>n. g(Suc n) \<le> g(n);
```
```   474          \<forall>n. f(n) \<le> g(n) |]
```
```   475       ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
```
```   476                             ((\<forall>n. m \<le> g(n)) & g ----> m)"
```
```   477 apply (subgoal_tac "monoseq f & monoseq g")
```
```   478 prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
```
```   479 apply (subgoal_tac "Bseq f & Bseq g")
```
```   480 prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
```
```   481 apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
```
```   482 apply (rule_tac x = "lim f" in exI)
```
```   483 apply (rule_tac x = "lim g" in exI)
```
```   484 apply (auto intro: LIMSEQ_le)
```
```   485 apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
```
```   486 done
```
```   487
```
```   488 lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
```
```   489          \<forall>n. g(Suc n) \<le> g(n);
```
```   490          \<forall>n. f(n) \<le> g(n);
```
```   491          (%n. f(n) - g(n)) ----> 0 |]
```
```   492       ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
```
```   493                 ((\<forall>n. l \<le> g(n)) & g ----> l)"
```
```   494 apply (drule lemma_nest, auto)
```
```   495 apply (subgoal_tac "l = m")
```
```   496 apply (drule_tac [2] X = f in LIMSEQ_diff)
```
```   497 apply (auto intro: LIMSEQ_unique)
```
```   498 done
```
```   499
```
```   500 text{*The universal quantifiers below are required for the declaration
```
```   501   of @{text Bolzano_nest_unique} below.*}
```
```   502
```
```   503 lemma Bolzano_bisect_le:
```
```   504  "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
```
```   505 apply (rule allI)
```
```   506 apply (induct_tac "n")
```
```   507 apply (auto simp add: Let_def split_def)
```
```   508 done
```
```   509
```
```   510 lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
```
```   511    \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
```
```   512 apply (rule allI)
```
```   513 apply (induct_tac "n")
```
```   514 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
```
```   515 done
```
```   516
```
```   517 lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
```
```   518    \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
```
```   519 apply (rule allI)
```
```   520 apply (induct_tac "n")
```
```   521 apply (auto simp add: Bolzano_bisect_le Let_def split_def)
```
```   522 done
```
```   523
```
```   524 lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
```
```   525 apply (auto)
```
```   526 apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
```
```   527 apply (simp)
```
```   528 done
```
```   529
```
```   530 lemma Bolzano_bisect_diff:
```
```   531      "a \<le> b ==>
```
```   532       snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
```
```   533       (b-a) / (2 ^ n)"
```
```   534 apply (induct "n")
```
```   535 apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
```
```   536 done
```
```   537
```
```   538 lemmas Bolzano_nest_unique =
```
```   539     lemma_nest_unique
```
```   540     [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
```
```   541
```
```   542
```
```   543 lemma not_P_Bolzano_bisect:
```
```   544   assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
```
```   545       and notP: "~ P(a,b)"
```
```   546       and le:   "a \<le> b"
```
```   547   shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
```
```   548 proof (induct n)
```
```   549   case 0 show ?case using notP by simp
```
```   550  next
```
```   551   case (Suc n)
```
```   552   thus ?case
```
```   553  by (auto simp del: surjective_pairing [symmetric]
```
```   554              simp add: Let_def split_def Bolzano_bisect_le [OF le]
```
```   555      P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
```
```   556 qed
```
```   557
```
```   558 (*Now we re-package P_prem as a formula*)
```
```   559 lemma not_P_Bolzano_bisect':
```
```   560      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
```
```   561          ~ P(a,b);  a \<le> b |] ==>
```
```   562       \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
```
```   563 by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
```
```   564
```
```   565
```
```   566
```
```   567 lemma lemma_BOLZANO:
```
```   568      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
```
```   569          \<forall>x. \<exists>d::real. 0 < d &
```
```   570                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
```
```   571          a \<le> b |]
```
```   572       ==> P(a,b)"
```
```   573 apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
```
```   574 apply (rule LIMSEQ_minus_cancel)
```
```   575 apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
```
```   576 apply (rule ccontr)
```
```   577 apply (drule not_P_Bolzano_bisect', assumption+)
```
```   578 apply (rename_tac "l")
```
```   579 apply (drule_tac x = l in spec, clarify)
```
```   580 apply (simp add: LIMSEQ_def)
```
```   581 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
```
```   582 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
```
```   583 apply (drule real_less_half_sum, auto)
```
```   584 apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
```
```   585 apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
```
```   586 apply safe
```
```   587 apply (simp_all (no_asm_simp))
```
```   588 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)
```
```   589 apply (simp (no_asm_simp) add: abs_if)
```
```   590 apply (rule real_sum_of_halves [THEN subst])
```
```   591 apply (rule add_strict_mono)
```
```   592 apply (simp_all add: diff_minus [symmetric])
```
```   593 done
```
```   594
```
```   595
```
```   596 lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
```
```   597        (\<forall>x. \<exists>d::real. 0 < d &
```
```   598                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
```
```   599       --> (\<forall>a b. a \<le> b --> P(a,b))"
```
```   600 apply clarify
```
```   601 apply (blast intro: lemma_BOLZANO)
```
```   602 done
```
```   603
```
```   604
```
```   605 subsection {* Intermediate Value Theorem *}
```
```   606
```
```   607 text {*Prove Contrapositive by Bisection*}
```
```   608
```
```   609 lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
```
```   610          a \<le> b;
```
```   611          (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
```
```   612       ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
```
```   613 apply (rule contrapos_pp, assumption)
```
```   614 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)
```
```   615 apply safe
```
```   616 apply simp_all
```
```   617 apply (simp add: isCont_iff LIM_def)
```
```   618 apply (rule ccontr)
```
```   619 apply (subgoal_tac "a \<le> x & x \<le> b")
```
```   620  prefer 2
```
```   621  apply simp
```
```   622  apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
```
```   623 apply (drule_tac x = x in spec)+
```
```   624 apply simp
```
```   625 apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
```
```   626 apply safe
```
```   627 apply simp
```
```   628 apply (drule_tac x = s in spec, clarify)
```
```   629 apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
```
```   630 apply (drule_tac x = "ba-x" in spec)
```
```   631 apply (simp_all add: abs_if)
```
```   632 apply (drule_tac x = "aa-x" in spec)
```
```   633 apply (case_tac "x \<le> aa", simp_all)
```
```   634 done
```
```   635
```
```   636 lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
```
```   637          a \<le> b;
```
```   638          (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
```
```   639       |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
```
```   640 apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
```
```   641 apply (drule IVT [where f = "%x. - f x"], assumption)
```
```   642 apply (auto intro: isCont_minus)
```
```   643 done
```
```   644
```
```   645 (*HOL style here: object-level formulations*)
```
```   646 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
```
```   647       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```   648       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```   649 apply (blast intro: IVT)
```
```   650 done
```
```   651
```
```   652 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
```
```   653       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```   654       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```   655 apply (blast intro: IVT2)
```
```   656 done
```
```   657
```
```   658 text{*By bisection, function continuous on closed interval is bounded above*}
```
```   659
```
```   660 lemma isCont_bounded:
```
```   661      "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   662       ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
```
```   663 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)
```
```   664 apply safe
```
```   665 apply simp_all
```
```   666 apply (rename_tac x xa ya M Ma)
```
```   667 apply (cut_tac x = M and y = Ma in linorder_linear, safe)
```
```   668 apply (rule_tac x = Ma in exI, clarify)
```
```   669 apply (cut_tac x = xb and y = xa in linorder_linear, force)
```
```   670 apply (rule_tac x = M in exI, clarify)
```
```   671 apply (cut_tac x = xb and y = xa in linorder_linear, force)
```
```   672 apply (case_tac "a \<le> x & x \<le> b")
```
```   673 apply (rule_tac [2] x = 1 in exI)
```
```   674 prefer 2 apply force
```
```   675 apply (simp add: LIM_def isCont_iff)
```
```   676 apply (drule_tac x = x in spec, auto)
```
```   677 apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
```
```   678 apply (drule_tac x = 1 in spec, auto)
```
```   679 apply (rule_tac x = s in exI, clarify)
```
```   680 apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
```
```   681 apply (drule_tac x = "xa-x" in spec)
```
```   682 apply (auto simp add: abs_ge_self)
```
```   683 done
```
```   684
```
```   685 text{*Refine the above to existence of least upper bound*}
```
```   686
```
```   687 lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
```
```   688       (\<exists>t. isLub UNIV S t)"
```
```   689 by (blast intro: reals_complete)
```
```   690
```
```   691 lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   692          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
```
```   693                    (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
```
```   694 apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
```
```   695         in lemma_reals_complete)
```
```   696 apply auto
```
```   697 apply (drule isCont_bounded, assumption)
```
```   698 apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
```
```   699 apply (rule exI, auto)
```
```   700 apply (auto dest!: spec simp add: linorder_not_less)
```
```   701 done
```
```   702
```
```   703 text{*Now show that it attains its upper bound*}
```
```   704
```
```   705 lemma isCont_eq_Ub:
```
```   706   assumes le: "a \<le> b"
```
```   707       and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
```
```   708   shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
```
```   709              (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
```
```   710 proof -
```
```   711   from isCont_has_Ub [OF le con]
```
```   712   obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
```
```   713              and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
```
```   714   show ?thesis
```
```   715   proof (intro exI, intro conjI)
```
```   716     show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
```
```   717     show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
```
```   718     proof (rule ccontr)
```
```   719       assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
```
```   720       with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
```
```   721         by (fastsimp simp add: linorder_not_le [symmetric])
```
```   722       hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
```
```   723         by (auto simp add: isCont_inverse isCont_diff con)
```
```   724       from isCont_bounded [OF le this]
```
```   725       obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
```
```   726       have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
```
```   727         by (simp add: M3 compare_rls)
```
```   728       have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
```
```   729         by (auto intro: order_le_less_trans [of _ k])
```
```   730       with Minv
```
```   731       have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
```
```   732         by (intro strip less_imp_inverse_less, simp_all)
```
```   733       hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
```
```   734         by simp
```
```   735       have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
```
```   736         by (simp, arith)
```
```   737       from M2 [OF this]
```
```   738       obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
```
```   739       thus False using invlt [of x] by force
```
```   740     qed
```
```   741   qed
```
```   742 qed
```
```   743
```
```   744
```
```   745 text{*Same theorem for lower bound*}
```
```   746
```
```   747 lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   748          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
```
```   749                    (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
```
```   750 apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
```
```   751 prefer 2 apply (blast intro: isCont_minus)
```
```   752 apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
```
```   753 apply safe
```
```   754 apply auto
```
```   755 done
```
```   756
```
```   757
```
```   758 text{*Another version.*}
```
```   759
```
```   760 lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
```
```   761       ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
```
```   762           (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
```
```   763 apply (frule isCont_eq_Lb)
```
```   764 apply (frule_tac [2] isCont_eq_Ub)
```
```   765 apply (assumption+, safe)
```
```   766 apply (rule_tac x = "f x" in exI)
```
```   767 apply (rule_tac x = "f xa" in exI, simp, safe)
```
```   768 apply (cut_tac x = x and y = xa in linorder_linear, safe)
```
```   769 apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
```
```   770 apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
```
```   771 apply (rule_tac [2] x = xb in exI)
```
```   772 apply (rule_tac [4] x = xb in exI, simp_all)
```
```   773 done
```
```   774
```
```   775
```
```   776 text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
```
```   777
```
```   778 lemma DERIV_left_inc:
```
```   779   fixes f :: "real => real"
```
```   780   assumes der: "DERIV f x :> l"
```
```   781       and l:   "0 < l"
```
```   782   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
```
```   783 proof -
```
```   784   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
```
```   785   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)"
```
```   786     by (simp add: diff_minus)
```
```   787   then obtain s
```
```   788         where s:   "0 < s"
```
```   789           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
```
```   790     by auto
```
```   791   thus ?thesis
```
```   792   proof (intro exI conjI strip)
```
```   793     show "0<s" using s .
```
```   794     fix h::real
```
```   795     assume "0 < h" "h < s"
```
```   796     with all [of h] show "f x < f (x+h)"
```
```   797     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   798     split add: split_if_asm)
```
```   799       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
```
```   800       with l
```
```   801       have "0 < (f (x+h) - f x) / h" by arith
```
```   802       thus "f x < f (x+h)"
```
```   803   by (simp add: pos_less_divide_eq h)
```
```   804     qed
```
```   805   qed
```
```   806 qed
```
```   807
```
```   808 lemma DERIV_left_dec:
```
```   809   fixes f :: "real => real"
```
```   810   assumes der: "DERIV f x :> l"
```
```   811       and l:   "l < 0"
```
```   812   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
```
```   813 proof -
```
```   814   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
```
```   815   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)"
```
```   816     by (simp add: diff_minus)
```
```   817   then obtain s
```
```   818         where s:   "0 < s"
```
```   819           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
```
```   820     by auto
```
```   821   thus ?thesis
```
```   822   proof (intro exI conjI strip)
```
```   823     show "0<s" using s .
```
```   824     fix h::real
```
```   825     assume "0 < h" "h < s"
```
```   826     with all [of "-h"] show "f x < f (x-h)"
```
```   827     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
```
```   828     split add: split_if_asm)
```
```   829       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
```
```   830       with l
```
```   831       have "0 < (f (x-h) - f x) / h" by arith
```
```   832       thus "f x < f (x-h)"
```
```   833   by (simp add: pos_less_divide_eq h)
```
```   834     qed
```
```   835   qed
```
```   836 qed
```
```   837
```
```   838 lemma DERIV_local_max:
```
```   839   fixes f :: "real => real"
```
```   840   assumes der: "DERIV f x :> l"
```
```   841       and d:   "0 < d"
```
```   842       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
```
```   843   shows "l = 0"
```
```   844 proof (cases rule: linorder_cases [of l 0])
```
```   845   case equal thus ?thesis .
```
```   846 next
```
```   847   case less
```
```   848   from DERIV_left_dec [OF der less]
```
```   849   obtain d' where d': "0 < d'"
```
```   850              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
```
```   851   from real_lbound_gt_zero [OF d d']
```
```   852   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   853   with lt le [THEN spec [where x="x-e"]]
```
```   854   show ?thesis by (auto simp add: abs_if)
```
```   855 next
```
```   856   case greater
```
```   857   from DERIV_left_inc [OF der greater]
```
```   858   obtain d' where d': "0 < d'"
```
```   859              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
```
```   860   from real_lbound_gt_zero [OF d d']
```
```   861   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
```
```   862   with lt le [THEN spec [where x="x+e"]]
```
```   863   show ?thesis by (auto simp add: abs_if)
```
```   864 qed
```
```   865
```
```   866
```
```   867 text{*Similar theorem for a local minimum*}
```
```   868 lemma DERIV_local_min:
```
```   869   fixes f :: "real => real"
```
```   870   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
```
```   871 by (drule DERIV_minus [THEN DERIV_local_max], auto)
```
```   872
```
```   873
```
```   874 text{*In particular, if a function is locally flat*}
```
```   875 lemma DERIV_local_const:
```
```   876   fixes f :: "real => real"
```
```   877   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
```
```   878 by (auto dest!: DERIV_local_max)
```
```   879
```
```   880 text{*Lemma about introducing open ball in open interval*}
```
```   881 lemma lemma_interval_lt:
```
```   882      "[| a < x;  x < b |]
```
```   883       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
```
```   884
```
```   885 apply (simp add: abs_less_iff)
```
```   886 apply (insert linorder_linear [of "x-a" "b-x"], safe)
```
```   887 apply (rule_tac x = "x-a" in exI)
```
```   888 apply (rule_tac [2] x = "b-x" in exI, auto)
```
```   889 done
```
```   890
```
```   891 lemma lemma_interval: "[| a < x;  x < b |] ==>
```
```   892         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
```
```   893 apply (drule lemma_interval_lt, auto)
```
```   894 apply (auto intro!: exI)
```
```   895 done
```
```   896
```
```   897 text{*Rolle's Theorem.
```
```   898    If @{term f} is defined and continuous on the closed interval
```
```   899    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
```
```   900    and @{term "f(a) = f(b)"},
```
```   901    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
```
```   902 theorem Rolle:
```
```   903   assumes lt: "a < b"
```
```   904       and eq: "f(a) = f(b)"
```
```   905       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```   906       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```   907   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
```
```   908 proof -
```
```   909   have le: "a \<le> b" using lt by simp
```
```   910   from isCont_eq_Ub [OF le con]
```
```   911   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
```
```   912              and alex: "a \<le> x" and xleb: "x \<le> b"
```
```   913     by blast
```
```   914   from isCont_eq_Lb [OF le con]
```
```   915   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
```
```   916               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
```
```   917     by blast
```
```   918   show ?thesis
```
```   919   proof cases
```
```   920     assume axb: "a < x & x < b"
```
```   921         --{*@{term f} attains its maximum within the interval*}
```
```   922     hence ax: "a<x" and xb: "x<b" by arith +
```
```   923     from lemma_interval [OF ax xb]
```
```   924     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   925       by blast
```
```   926     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
```
```   927       by blast
```
```   928     from differentiableD [OF dif [OF axb]]
```
```   929     obtain l where der: "DERIV f x :> l" ..
```
```   930     have "l=0" by (rule DERIV_local_max [OF der d bound'])
```
```   931         --{*the derivative at a local maximum is zero*}
```
```   932     thus ?thesis using ax xb der by auto
```
```   933   next
```
```   934     assume notaxb: "~ (a < x & x < b)"
```
```   935     hence xeqab: "x=a | x=b" using alex xleb by arith
```
```   936     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
```
```   937     show ?thesis
```
```   938     proof cases
```
```   939       assume ax'b: "a < x' & x' < b"
```
```   940         --{*@{term f} attains its minimum within the interval*}
```
```   941       hence ax': "a<x'" and x'b: "x'<b" by arith+
```
```   942       from lemma_interval [OF ax' x'b]
```
```   943       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   944   by blast
```
```   945       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
```
```   946   by blast
```
```   947       from differentiableD [OF dif [OF ax'b]]
```
```   948       obtain l where der: "DERIV f x' :> l" ..
```
```   949       have "l=0" by (rule DERIV_local_min [OF der d bound'])
```
```   950         --{*the derivative at a local minimum is zero*}
```
```   951       thus ?thesis using ax' x'b der by auto
```
```   952     next
```
```   953       assume notax'b: "~ (a < x' & x' < b)"
```
```   954         --{*@{term f} is constant througout the interval*}
```
```   955       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
```
```   956       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
```
```   957       from dense [OF lt]
```
```   958       obtain r where ar: "a < r" and rb: "r < b" by blast
```
```   959       from lemma_interval [OF ar rb]
```
```   960       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
```
```   961   by blast
```
```   962       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
```
```   963       proof (clarify)
```
```   964         fix z::real
```
```   965         assume az: "a \<le> z" and zb: "z \<le> b"
```
```   966         show "f z = f b"
```
```   967         proof (rule order_antisym)
```
```   968           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
```
```   969           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
```
```   970         qed
```
```   971       qed
```
```   972       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
```
```   973       proof (intro strip)
```
```   974         fix y::real
```
```   975         assume lt: "\<bar>r-y\<bar> < d"
```
```   976         hence "f y = f b" by (simp add: eq_fb bound)
```
```   977         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
```
```   978       qed
```
```   979       from differentiableD [OF dif [OF conjI [OF ar rb]]]
```
```   980       obtain l where der: "DERIV f r :> l" ..
```
```   981       have "l=0" by (rule DERIV_local_const [OF der d bound'])
```
```   982         --{*the derivative of a constant function is zero*}
```
```   983       thus ?thesis using ar rb der by auto
```
```   984     qed
```
```   985   qed
```
```   986 qed
```
```   987
```
```   988
```
```   989 subsection{*Mean Value Theorem*}
```
```   990
```
```   991 lemma lemma_MVT:
```
```   992      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
```
```   993 proof cases
```
```   994   assume "a=b" thus ?thesis by simp
```
```   995 next
```
```   996   assume "a\<noteq>b"
```
```   997   hence ba: "b-a \<noteq> 0" by arith
```
```   998   show ?thesis
```
```   999     by (rule real_mult_left_cancel [OF ba, THEN iffD1],
```
```  1000         simp add: right_diff_distrib,
```
```  1001         simp add: left_diff_distrib)
```
```  1002 qed
```
```  1003
```
```  1004 theorem MVT:
```
```  1005   assumes lt:  "a < b"
```
```  1006       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
```
```  1007       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
```
```  1008   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
```
```  1009                    (f(b) - f(a) = (b-a) * l)"
```
```  1010 proof -
```
```  1011   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
```
```  1012   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
```
```  1013     by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
```
```  1014   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
```
```  1015   proof (clarify)
```
```  1016     fix x::real
```
```  1017     assume ax: "a < x" and xb: "x < b"
```
```  1018     from differentiableD [OF dif [OF conjI [OF ax xb]]]
```
```  1019     obtain l where der: "DERIV f x :> l" ..
```
```  1020     show "?F differentiable x"
```
```  1021       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
```
```  1022           blast intro: DERIV_diff DERIV_cmult_Id der)
```
```  1023   qed
```
```  1024   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
```
```  1025   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
```
```  1026     by blast
```
```  1027   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
```
```  1028     by (rule DERIV_cmult_Id)
```
```  1029   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
```
```  1030                    :> 0 + (f b - f a) / (b - a)"
```
```  1031     by (rule DERIV_add [OF der])
```
```  1032   show ?thesis
```
```  1033   proof (intro exI conjI)
```
```  1034     show "a < z" using az .
```
```  1035     show "z < b" using zb .
```
```  1036     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
```
```  1037     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
```
```  1038   qed
```
```  1039 qed
```
```  1040
```
```  1041
```
```  1042 text{*A function is constant if its derivative is 0 over an interval.*}
```
```  1043
```
```  1044 lemma DERIV_isconst_end:
```
```  1045   fixes f :: "real => real"
```
```  1046   shows "[| a < b;
```
```  1047          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1048          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1049         ==> f b = f a"
```
```  1050 apply (drule MVT, assumption)
```
```  1051 apply (blast intro: differentiableI)
```
```  1052 apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
```
```  1053 done
```
```  1054
```
```  1055 lemma DERIV_isconst1:
```
```  1056   fixes f :: "real => real"
```
```  1057   shows "[| a < b;
```
```  1058          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1059          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
```
```  1060         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
```
```  1061 apply safe
```
```  1062 apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
```
```  1063 apply (drule_tac b = x in DERIV_isconst_end, auto)
```
```  1064 done
```
```  1065
```
```  1066 lemma DERIV_isconst2:
```
```  1067   fixes f :: "real => real"
```
```  1068   shows "[| a < b;
```
```  1069          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
```
```  1070          \<forall>x. a < x & x < b --> DERIV f x :> 0;
```
```  1071          a \<le> x; x \<le> b |]
```
```  1072         ==> f x = f a"
```
```  1073 apply (blast dest: DERIV_isconst1)
```
```  1074 done
```
```  1075
```
```  1076 lemma DERIV_isconst_all:
```
```  1077   fixes f :: "real => real"
```
```  1078   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
```
```  1079 apply (rule linorder_cases [of x y])
```
```  1080 apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
```
```  1081 done
```
```  1082
```
```  1083 lemma DERIV_const_ratio_const:
```
```  1084   fixes f :: "real => real"
```
```  1085   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
```
```  1086 apply (rule linorder_cases [of a b], auto)
```
```  1087 apply (drule_tac [!] f = f in MVT)
```
```  1088 apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
```
```  1089 apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
```
```  1090 done
```
```  1091
```
```  1092 lemma DERIV_const_ratio_const2:
```
```  1093   fixes f :: "real => real"
```
```  1094   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
```
```  1095 apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
```
```  1096 apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
```
```  1097 done
```
```  1098
```
```  1099 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
```
```  1100 by (simp)
```
```  1101
```
```  1102 lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
```
```  1103 by (simp)
```
```  1104
```
```  1105 text{*Gallileo's "trick": average velocity = av. of end velocities*}
```
```  1106
```
```  1107 lemma DERIV_const_average:
```
```  1108   fixes v :: "real => real"
```
```  1109   assumes neq: "a \<noteq> (b::real)"
```
```  1110       and der: "\<forall>x. DERIV v x :> k"
```
```  1111   shows "v ((a + b)/2) = (v a + v b)/2"
```
```  1112 proof (cases rule: linorder_cases [of a b])
```
```  1113   case equal with neq show ?thesis by simp
```
```  1114 next
```
```  1115   case less
```
```  1116   have "(v b - v a) / (b - a) = k"
```
```  1117     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1118   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1119   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
```
```  1120     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1121   ultimately show ?thesis using neq by force
```
```  1122 next
```
```  1123   case greater
```
```  1124   have "(v b - v a) / (b - a) = k"
```
```  1125     by (rule DERIV_const_ratio_const2 [OF neq der])
```
```  1126   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
```
```  1127   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
```
```  1128     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
```
```  1129   ultimately show ?thesis using neq by (force simp add: add_commute)
```
```  1130 qed
```
```  1131
```
```  1132
```
```  1133 text{*Dull lemma: an continuous injection on an interval must have a
```
```  1134 strict maximum at an end point, not in the middle.*}
```
```  1135
```
```  1136 lemma lemma_isCont_inj:
```
```  1137   fixes f :: "real \<Rightarrow> real"
```
```  1138   assumes d: "0 < d"
```
```  1139       and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1140       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1141   shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
```
```  1142 proof (rule ccontr)
```
```  1143   assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
```
```  1144   hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
```
```  1145   show False
```
```  1146   proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
```
```  1147     case le
```
```  1148     from d cont all [of "x+d"]
```
```  1149     have flef: "f(x+d) \<le> f x"
```
```  1150      and xlex: "x - d \<le> x"
```
```  1151      and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
```
```  1152        by (auto simp add: abs_if)
```
```  1153     from IVT [OF le flef xlex cont']
```
```  1154     obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
```
```  1155     moreover
```
```  1156     hence "g(f x') = g (f(x+d))" by simp
```
```  1157     ultimately show False using d inj [of x'] inj [of "x+d"]
```
```  1158       by (simp add: abs_le_iff)
```
```  1159   next
```
```  1160     case ge
```
```  1161     from d cont all [of "x-d"]
```
```  1162     have flef: "f(x-d) \<le> f x"
```
```  1163      and xlex: "x \<le> x+d"
```
```  1164      and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
```
```  1165        by (auto simp add: abs_if)
```
```  1166     from IVT2 [OF ge flef xlex cont']
```
```  1167     obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
```
```  1168     moreover
```
```  1169     hence "g(f x') = g (f(x-d))" by simp
```
```  1170     ultimately show False using d inj [of x'] inj [of "x-d"]
```
```  1171       by (simp add: abs_le_iff)
```
```  1172   qed
```
```  1173 qed
```
```  1174
```
```  1175
```
```  1176 text{*Similar version for lower bound.*}
```
```  1177
```
```  1178 lemma lemma_isCont_inj2:
```
```  1179   fixes f g :: "real \<Rightarrow> real"
```
```  1180   shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
```
```  1181         \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
```
```  1182       ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
```
```  1183 apply (insert lemma_isCont_inj
```
```  1184           [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
```
```  1185 apply (simp add: isCont_minus linorder_not_le)
```
```  1186 done
```
```  1187
```
```  1188 text{*Show there's an interval surrounding @{term "f(x)"} in
```
```  1189 @{text "f[[x - d, x + d]]"} .*}
```
```  1190
```
```  1191 lemma isCont_inj_range:
```
```  1192   fixes f :: "real \<Rightarrow> real"
```
```  1193   assumes d: "0 < d"
```
```  1194       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1195       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1196   shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
```
```  1197 proof -
```
```  1198   have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
```
```  1199     by (auto simp add: abs_le_iff)
```
```  1200   from isCont_Lb_Ub [OF this]
```
```  1201   obtain L M
```
```  1202   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"
```
```  1203     and all2 [rule_format]:
```
```  1204            "\<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)"
```
```  1205     by auto
```
```  1206   with d have "L \<le> f x & f x \<le> M" by simp
```
```  1207   moreover have "L \<noteq> f x"
```
```  1208   proof -
```
```  1209     from lemma_isCont_inj2 [OF d inj cont]
```
```  1210     obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
```
```  1211     thus ?thesis using all1 [of u] by arith
```
```  1212   qed
```
```  1213   moreover have "f x \<noteq> M"
```
```  1214   proof -
```
```  1215     from lemma_isCont_inj [OF d inj cont]
```
```  1216     obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
```
```  1217     thus ?thesis using all1 [of u] by arith
```
```  1218   qed
```
```  1219   ultimately have "L < f x & f x < M" by arith
```
```  1220   hence "0 < f x - L" "0 < M - f x" by arith+
```
```  1221   from real_lbound_gt_zero [OF this]
```
```  1222   obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
```
```  1223   thus ?thesis
```
```  1224   proof (intro exI conjI)
```
```  1225     show "0<e" using e(1) .
```
```  1226     show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
```
```  1227     proof (intro strip)
```
```  1228       fix y::real
```
```  1229       assume "\<bar>y - f x\<bar> \<le> e"
```
```  1230       with e have "L \<le> y \<and> y \<le> M" by arith
```
```  1231       from all2 [OF this]
```
```  1232       obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
```
```  1233       thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
```
```  1234         by (force simp add: abs_le_iff)
```
```  1235     qed
```
```  1236   qed
```
```  1237 qed
```
```  1238
```
```  1239
```
```  1240 text{*Continuity of inverse function*}
```
```  1241
```
```  1242 lemma isCont_inverse_function:
```
```  1243   fixes f g :: "real \<Rightarrow> real"
```
```  1244   assumes d: "0 < d"
```
```  1245       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
```
```  1246       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
```
```  1247   shows "isCont g (f x)"
```
```  1248 proof (simp add: isCont_iff LIM_eq)
```
```  1249   show "\<forall>r. 0 < r \<longrightarrow>
```
```  1250          (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
```
```  1251   proof (intro strip)
```
```  1252     fix r::real
```
```  1253     assume r: "0<r"
```
```  1254     from real_lbound_gt_zero [OF r d]
```
```  1255     obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
```
```  1256     with inj cont
```
```  1257     have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
```
```  1258                   "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
```
```  1259     from isCont_inj_range [OF e this]
```
```  1260     obtain e' where e': "0 < e'"
```
```  1261         and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
```
```  1262           by blast
```
```  1263     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"
```
```  1264     proof (intro exI conjI)
```
```  1265       show "0<e'" using e' .
```
```  1266       show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
```
```  1267       proof (intro strip)
```
```  1268         fix z::real
```
```  1269         assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
```
```  1270         with e e_lt e_simps all [rule_format, of "f x + z"]
```
```  1271         show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
```
```  1272       qed
```
```  1273     qed
```
```  1274   qed
```
```  1275 qed
```
```  1276
```
```  1277 text {* Derivative of inverse function *}
```
```  1278
```
```  1279 lemma DERIV_inverse_function:
```
```  1280   fixes f g :: "real \<Rightarrow> real"
```
```  1281   assumes der: "DERIV f (g x) :> D"
```
```  1282   assumes neq: "D \<noteq> 0"
```
```  1283   assumes a: "a < x" and b: "x < b"
```
```  1284   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
```
```  1285   assumes cont: "isCont g x"
```
```  1286   shows "DERIV g x :> inverse D"
```
```  1287 unfolding DERIV_iff2
```
```  1288 proof (rule LIM_equal2)
```
```  1289   show "0 < min (x - a) (b - x)"
```
```  1290     using a b by arith
```
```  1291 next
```
```  1292   fix y
```
```  1293   assume "norm (y - x) < min (x - a) (b - x)"
```
```  1294   hence "a < y" and "y < b"
```
```  1295     by (simp_all add: abs_less_iff)
```
```  1296   thus "(g y - g x) / (y - x) =
```
```  1297         inverse ((f (g y) - x) / (g y - g x))"
```
```  1298     by (simp add: inj)
```
```  1299 next
```
```  1300   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
```
```  1301     by (rule der [unfolded DERIV_iff2])
```
```  1302   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
```
```  1303     using inj a b by simp
```
```  1304   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
```
```  1305   proof (safe intro!: exI)
```
```  1306     show "0 < min (x - a) (b - x)"
```
```  1307       using a b by simp
```
```  1308   next
```
```  1309     fix y
```
```  1310     assume "norm (y - x) < min (x - a) (b - x)"
```
```  1311     hence y: "a < y" "y < b"
```
```  1312       by (simp_all add: abs_less_iff)
```
```  1313     assume "g y = g x"
```
```  1314     hence "f (g y) = f (g x)" by simp
```
```  1315     hence "y = x" using inj y a b by simp
```
```  1316     also assume "y \<noteq> x"
```
```  1317     finally show False by simp
```
```  1318   qed
```
```  1319   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
```
```  1320     using cont 1 2 by (rule isCont_LIM_compose2)
```
```  1321   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
```
```  1322         -- x --> inverse D"
```
```  1323     using neq by (rule LIM_inverse)
```
```  1324 qed
```
```  1325
```
```  1326 theorem GMVT:
```
```  1327   fixes a b :: real
```
```  1328   assumes alb: "a < b"
```
```  1329   and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  1330   and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
```
```  1331   and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
```
```  1332   and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
```
```  1333   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)"
```
```  1334 proof -
```
```  1335   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
```
```  1336   from prems have "a < b" by simp
```
```  1337   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
```
```  1338   proof -
```
```  1339     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
```
```  1340     with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
```
```  1341       by (auto intro: isCont_mult)
```
```  1342     moreover
```
```  1343     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
```
```  1344     with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
```
```  1345       by (auto intro: isCont_mult)
```
```  1346     ultimately show ?thesis
```
```  1347       by (fastsimp intro: isCont_diff)
```
```  1348   qed
```
```  1349   moreover
```
```  1350   have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
```
```  1351   proof -
```
```  1352     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
```
```  1353     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)
```
```  1354     moreover
```
```  1355     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
```
```  1356     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)
```
```  1357     ultimately show ?thesis by (simp add: differentiable_diff)
```
```  1358   qed
```
```  1359   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)
```
```  1360   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" ..
```
```  1361   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
```
```  1362
```
```  1363   from cdef have cint: "a < c \<and> c < b" by auto
```
```  1364   with gd have "g differentiable c" by simp
```
```  1365   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
```
```  1366   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
```
```  1367
```
```  1368   from cdef have "a < c \<and> c < b" by auto
```
```  1369   with fd have "f differentiable c" by simp
```
```  1370   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
```
```  1371   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
```
```  1372
```
```  1373   from cdef have "DERIV ?h c :> l" by auto
```
```  1374   moreover
```
```  1375   {
```
```  1376     have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
```
```  1377       apply (insert DERIV_const [where k="f b - f a"])
```
```  1378       apply (drule meta_spec [of _ c])
```
```  1379       apply (drule DERIV_mult [OF _ g'cdef])
```
```  1380       by simp
```
```  1381     moreover have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
```
```  1382       apply (insert DERIV_const [where k="g b - g a"])
```
```  1383       apply (drule meta_spec [of _ c])
```
```  1384       apply (drule DERIV_mult [OF _ f'cdef])
```
```  1385       by simp
```
```  1386     ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
```
```  1387       by (simp add: DERIV_diff)
```
```  1388   }
```
```  1389   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
```
```  1390
```
```  1391   {
```
```  1392     from cdef have "?h b - ?h a = (b - a) * l" by auto
```
```  1393     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1394     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
```
```  1395   }
```
```  1396   moreover
```
```  1397   {
```
```  1398     have "?h b - ?h a =
```
```  1399          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
```
```  1400           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
```
```  1401       by (simp add: mult_ac add_ac right_diff_distrib)
```
```  1402     hence "?h b - ?h a = 0" by auto
```
```  1403   }
```
```  1404   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
```
```  1405   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
```
```  1406   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
```
```  1407   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
```
```  1408
```
```  1409   with g'cdef f'cdef cint show ?thesis by auto
```
```  1410 qed
```
```  1411
```
```  1412 lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
```
```  1413 by auto
```
```  1414
```
```  1415
```
```  1416 subsection {* Derivatives of univariate polynomials *}
```
```  1417
```
```  1418 definition
```
```  1419   pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly" where
```
```  1420   "pderiv = poly_rec 0 (\<lambda>a p p'. p + pCons 0 p')"
```
```  1421
```
```  1422 lemma pderiv_0 [simp]: "pderiv 0 = 0"
```
```  1423   unfolding pderiv_def by (simp add: poly_rec_0)
```
```  1424
```
```  1425 lemma pderiv_pCons: "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
```
```  1426   unfolding pderiv_def by (simp add: poly_rec_pCons)
```
```  1427
```
```  1428 lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
```
```  1429   apply (induct p arbitrary: n, simp)
```
```  1430   apply (simp add: pderiv_pCons coeff_pCons ring_simps split: nat.split)
```
```  1431   done
```
```  1432
```
```  1433 lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
```
```  1434   apply (rule iffI)
```
```  1435   apply (cases p, simp)
```
```  1436   apply (simp add: expand_poly_eq coeff_pderiv del: of_nat_Suc)
```
```  1437   apply (simp add: expand_poly_eq coeff_pderiv coeff_eq_0)
```
```  1438   done
```
```  1439
```
```  1440 lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
```
```  1441   apply (rule order_antisym [OF degree_le])
```
```  1442   apply (simp add: coeff_pderiv coeff_eq_0)
```
```  1443   apply (cases "degree p", simp)
```
```  1444   apply (rule le_degree)
```
```  1445   apply (simp add: coeff_pderiv del: of_nat_Suc)
```
```  1446   apply (rule subst, assumption)
```
```  1447   apply (rule leading_coeff_neq_0, clarsimp)
```
```  1448   done
```
```  1449
```
```  1450 lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
```
```  1451 by (simp add: pderiv_pCons)
```
```  1452
```
```  1453 lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
```
```  1454 by (rule poly_ext, simp add: coeff_pderiv ring_simps)
```
```  1455
```
```  1456 lemma pderiv_minus: "pderiv (- p) = - pderiv p"
```
```  1457 by (rule poly_ext, simp add: coeff_pderiv)
```
```  1458
```
```  1459 lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
```
```  1460 by (rule poly_ext, simp add: coeff_pderiv ring_simps)
```
```  1461
```
```  1462 lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
```
```  1463 by (rule poly_ext, simp add: coeff_pderiv ring_simps)
```
```  1464
```
```  1465 lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
```
```  1466 apply (induct p)
```
```  1467 apply simp
```
```  1468 apply (simp add: pderiv_add pderiv_smult pderiv_pCons ring_simps)
```
```  1469 done
```
```  1470
```
```  1471 lemma pderiv_power_Suc:
```
```  1472   "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
```
```  1473 apply (induct n)
```
```  1474 apply simp
```
```  1475 apply (subst power_Suc)
```
```  1476 apply (subst pderiv_mult)
```
```  1477 apply (erule ssubst)
```
```  1478 apply (simp add: mult_smult_right mult_smult_left smult_add_left ring_simps)
```
```  1479 done
```
```  1480
```
```  1481 lemma DERIV_cmult2: "DERIV f x :> D ==> DERIV (%x. (f x) * c :: real) x :> D * c"
```
```  1482 by (simp add: DERIV_cmult mult_commute [of _ c])
```
```  1483
```
```  1484 lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
```
```  1485 by (rule lemma_DERIV_subst, rule DERIV_pow, simp)
```
```  1486 declare DERIV_pow2 [simp] DERIV_pow [simp]
```
```  1487
```
```  1488 lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
```
```  1489 by (rule lemma_DERIV_subst, rule DERIV_add, auto)
```
```  1490
```
```  1491 lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
```
```  1492 apply (induct p)
```
```  1493 apply simp
```
```  1494 apply (simp add: pderiv_pCons)
```
```  1495 apply (rule lemma_DERIV_subst)
```
```  1496 apply (rule DERIV_add DERIV_mult DERIV_const DERIV_ident | assumption)+
```
```  1497 apply simp
```
```  1498 done
```
```  1499
```
```  1500 text{* Consequences of the derivative theorem above*}
```
```  1501
```
```  1502 lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (x::real)"
```
```  1503 apply (simp add: differentiable_def)
```
```  1504 apply (blast intro: poly_DERIV)
```
```  1505 done
```
```  1506
```
```  1507 lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
```
```  1508 by (rule poly_DERIV [THEN DERIV_isCont])
```
```  1509
```
```  1510 lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
```
```  1511       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
```
```  1512 apply (cut_tac f = "%x. poly p x" and a = a and b = b and y = 0 in IVT_objl)
```
```  1513 apply (auto simp add: order_le_less)
```
```  1514 done
```
```  1515
```
```  1516 lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
```
```  1517       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
```
```  1518 by (insert poly_IVT_pos [where p = "- p" ]) simp
```
```  1519
```
```  1520 lemma poly_MVT: "(a::real) < b ==>
```
```  1521      \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
```
```  1522 apply (drule_tac f = "poly p" in MVT, auto)
```
```  1523 apply (rule_tac x = z in exI)
```
```  1524 apply (auto simp add: real_mult_left_cancel poly_DERIV [THEN DERIV_unique])
```
```  1525 done
```
```  1526
```
```  1527 text{*Lemmas for Derivatives*}
```
```  1528
```
```  1529 (* FIXME
```
```  1530 lemma lemma_order_pderiv [rule_format]:
```
```  1531      "\<forall>p q a. 0 < n &
```
```  1532        poly (pderiv p) \<noteq> poly [] &
```
```  1533        poly p = poly ([- a, 1] %^ n *** q) & ~ [- a, 1] divides q
```
```  1534        --> n = Suc (order a (pderiv p))"
```
```  1535 apply (induct "n", safe)
```
```  1536 apply (rule order_unique_lemma, rule conjI, assumption)
```
```  1537 apply (subgoal_tac "\<forall>r. r divides (pderiv p) = r divides (pderiv ([-a, 1] %^ Suc n *** q))")
```
```  1538 apply (drule_tac [2] poly_pderiv_welldef)
```
```  1539  prefer 2 apply (simp add: divides_def del: pmult_Cons pexp_Suc)
```
```  1540 apply (simp del: pmult_Cons pexp_Suc)
```
```  1541 apply (rule conjI)
```
```  1542 apply (simp add: divides_def fun_eq del: pmult_Cons pexp_Suc)
```
```  1543 apply (rule_tac x = "[-a, 1] *** (pderiv q) +++ real (Suc n) %* q" in exI)
```
```  1544 apply (simp add: poly_pderiv_mult poly_pderiv_exp_prime poly_add poly_mult poly_cmult right_distrib mult_ac del: pmult_Cons pexp_Suc)
```
```  1545 apply (simp add: poly_mult right_distrib left_distrib mult_ac del: pmult_Cons)
```
```  1546 apply (erule_tac V = "\<forall>r. r divides pderiv p = r divides pderiv ([- a, 1] %^ Suc n *** q)" in thin_rl)
```
```  1547 apply (unfold divides_def)
```
```  1548 apply (simp (no_asm) add: poly_pderiv_mult poly_pderiv_exp_prime fun_eq poly_add poly_mult del: pmult_Cons pexp_Suc)
```
```  1549 apply (rule contrapos_np, assumption)
```
```  1550 apply (rotate_tac 3, erule contrapos_np)
```
```  1551 apply (simp del: pmult_Cons pexp_Suc, safe)
```
```  1552 apply (rule_tac x = "inverse (real (Suc n)) %* (qa +++ -- (pderiv q))" in exI)
```
```  1553 apply (subgoal_tac "poly ([-a, 1] %^ n *** q) = poly ([-a, 1] %^ n *** ([-a, 1] *** (inverse (real (Suc n)) %* (qa +++ -- (pderiv q))))) ")
```
```  1554 apply (drule poly_mult_left_cancel [THEN iffD1], simp)
```
```  1555 apply (simp add: fun_eq poly_mult poly_add poly_cmult poly_minus del: pmult_Cons mult_cancel_left, safe)
```
```  1556 apply (rule_tac c1 = "real (Suc n)" in real_mult_left_cancel [THEN iffD1])
```
```  1557 apply (simp (no_asm))
```
```  1558 apply (subgoal_tac "real (Suc n) * (poly ([- a, 1] %^ n) xa * poly q xa) =
```
```  1559           (poly qa xa + - poly (pderiv q) xa) *
```
```  1560           (poly ([- a, 1] %^ n) xa *
```
```  1561            ((- a + xa) * (inverse (real (Suc n)) * real (Suc n))))")
```
```  1562 apply (simp only: mult_ac)
```
```  1563 apply (rotate_tac 2)
```
```  1564 apply (drule_tac x = xa in spec)
```
```  1565 apply (simp add: left_distrib mult_ac del: pmult_Cons)
```
```  1566 done
```
```  1567
```
```  1568 lemma order_pderiv: "[| poly (pderiv p) \<noteq> poly []; order a p \<noteq> 0 |]
```
```  1569       ==> (order a p = Suc (order a (pderiv p)))"
```
```  1570 apply (case_tac "poly p = poly []")
```
```  1571 apply (auto dest: pderiv_zero)
```
```  1572 apply (drule_tac a = a and p = p in order_decomp)
```
```  1573 using neq0_conv
```
```  1574 apply (blast intro: lemma_order_pderiv)
```
```  1575 done
```
```  1576
```
```  1577 text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *}
```
```  1578
```
```  1579 lemma poly_squarefree_decomp_order: "[| poly (pderiv p) \<noteq> poly [];
```
```  1580          poly p = poly (q *** d);
```
```  1581          poly (pderiv p) = poly (e *** d);
```
```  1582          poly d = poly (r *** p +++ s *** pderiv p)
```
```  1583       |] ==> order a q = (if order a p = 0 then 0 else 1)"
```
```  1584 apply (subgoal_tac "order a p = order a q + order a d")
```
```  1585 apply (rule_tac [2] s = "order a (q *** d)" in trans)
```
```  1586 prefer 2 apply (blast intro: order_poly)
```
```  1587 apply (rule_tac [2] order_mult)
```
```  1588  prefer 2 apply force
```
```  1589 apply (case_tac "order a p = 0", simp)
```
```  1590 apply (subgoal_tac "order a (pderiv p) = order a e + order a d")
```
```  1591 apply (rule_tac [2] s = "order a (e *** d)" in trans)
```
```  1592 prefer 2 apply (blast intro: order_poly)
```
```  1593 apply (rule_tac [2] order_mult)
```
```  1594  prefer 2 apply force
```
```  1595 apply (case_tac "poly p = poly []")
```
```  1596 apply (drule_tac p = p in pderiv_zero, simp)
```
```  1597 apply (drule order_pderiv, assumption)
```
```  1598 apply (subgoal_tac "order a (pderiv p) \<le> order a d")
```
```  1599 apply (subgoal_tac [2] " ([-a, 1] %^ (order a (pderiv p))) divides d")
```
```  1600  prefer 2 apply (simp add: poly_entire order_divides)
```
```  1601 apply (subgoal_tac [2] " ([-a, 1] %^ (order a (pderiv p))) divides p & ([-a, 1] %^ (order a (pderiv p))) divides (pderiv p) ")
```
```  1602  prefer 3 apply (simp add: order_divides)
```
```  1603  prefer 2 apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
```
```  1604 apply (rule_tac x = "r *** qa +++ s *** qaa" in exI)
```
```  1605 apply (simp add: fun_eq poly_add poly_mult left_distrib right_distrib mult_ac del: pexp_Suc pmult_Cons, auto)
```
```  1606 done
```
```  1607
```
```  1608
```
```  1609 lemma poly_squarefree_decomp_order2: "[| poly (pderiv p) \<noteq> poly [];
```
```  1610          poly p = poly (q *** d);
```
```  1611          poly (pderiv p) = poly (e *** d);
```
```  1612          poly d = poly (r *** p +++ s *** pderiv p)
```
```  1613       |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
```
```  1614 apply (blast intro: poly_squarefree_decomp_order)
```
```  1615 done
```
```  1616
```
```  1617 lemma order_pderiv2: "[| poly (pderiv p) \<noteq> poly []; order a p \<noteq> 0 |]
```
```  1618       ==> (order a (pderiv p) = n) = (order a p = Suc n)"
```
```  1619 apply (auto dest: order_pderiv)
```
```  1620 done
```
```  1621
```
```  1622 lemma rsquarefree_roots:
```
```  1623   "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
```
```  1624 apply (simp add: rsquarefree_def)
```
```  1625 apply (case_tac "poly p = poly []", simp, simp)
```
```  1626 apply (case_tac "poly (pderiv p) = poly []")
```
```  1627 apply simp
```
```  1628 apply (drule pderiv_iszero, clarify)
```
```  1629 apply (subgoal_tac "\<forall>a. order a p = order a [h]")
```
```  1630 apply (simp add: fun_eq)
```
```  1631 apply (rule allI)
```
```  1632 apply (cut_tac p = "[h]" and a = a in order_root)
```
```  1633 apply (simp add: fun_eq)
```
```  1634 apply (blast intro: order_poly)
```
```  1635 apply (auto simp add: order_root order_pderiv2)
```
```  1636 apply (erule_tac x="a" in allE, simp)
```
```  1637 done
```
```  1638
```
```  1639 lemma poly_squarefree_decomp: "[| poly (pderiv p) \<noteq> poly [];
```
```  1640          poly p = poly (q *** d);
```
```  1641          poly (pderiv p) = poly (e *** d);
```
```  1642          poly d = poly (r *** p +++ s *** pderiv p)
```
```  1643       |] ==> rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
```
```  1644 apply (frule poly_squarefree_decomp_order2, assumption+)
```
```  1645 apply (case_tac "poly p = poly []")
```
```  1646 apply (blast dest: pderiv_zero)
```
```  1647 apply (simp (no_asm) add: rsquarefree_def order_root del: pmult_Cons)
```
```  1648 apply (simp add: poly_entire del: pmult_Cons)
```
```  1649 done
```
```  1650 *)
```
```  1651
```
```  1652 subsection {* Theorems about Limits *}
```
```  1653
```
```  1654 (* need to rename second isCont_inverse *)
```
```  1655
```
```  1656 lemma isCont_inv_fun:
```
```  1657   fixes f g :: "real \<Rightarrow> real"
```
```  1658   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  1659          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  1660       ==> isCont g (f x)"
```
```  1661 by (rule isCont_inverse_function)
```
```  1662
```
```  1663 lemma isCont_inv_fun_inv:
```
```  1664   fixes f g :: "real \<Rightarrow> real"
```
```  1665   shows "[| 0 < d;
```
```  1666          \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  1667          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  1668        ==> \<exists>e. 0 < e &
```
```  1669              (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
```
```  1670 apply (drule isCont_inj_range)
```
```  1671 prefer 2 apply (assumption, assumption, auto)
```
```  1672 apply (rule_tac x = e in exI, auto)
```
```  1673 apply (rotate_tac 2)
```
```  1674 apply (drule_tac x = y in spec, auto)
```
```  1675 done
```
```  1676
```
```  1677
```
```  1678 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
```
```  1679 lemma LIM_fun_gt_zero:
```
```  1680      "[| f -- c --> (l::real); 0 < l |]
```
```  1681          ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
```
```  1682 apply (auto simp add: LIM_def)
```
```  1683 apply (drule_tac x = "l/2" in spec, safe, force)
```
```  1684 apply (rule_tac x = s in exI)
```
```  1685 apply (auto simp only: abs_less_iff)
```
```  1686 done
```
```  1687
```
```  1688 lemma LIM_fun_less_zero:
```
```  1689      "[| f -- c --> (l::real); l < 0 |]
```
```  1690       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
```
```  1691 apply (auto simp add: LIM_def)
```
```  1692 apply (drule_tac x = "-l/2" in spec, safe, force)
```
```  1693 apply (rule_tac x = s in exI)
```
```  1694 apply (auto simp only: abs_less_iff)
```
```  1695 done
```
```  1696
```
```  1697
```
```  1698 lemma LIM_fun_not_zero:
```
```  1699      "[| f -- c --> (l::real); l \<noteq> 0 |]
```
```  1700       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
```
```  1701 apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
```
```  1702 apply (drule LIM_fun_less_zero)
```
```  1703 apply (drule_tac [3] LIM_fun_gt_zero)
```
```  1704 apply force+
```
```  1705 done
```
```  1706
```
```  1707 end
```