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permissions  rwrr 
21164  1 
(* Title : Deriv.thy 
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ID : $Id$ 

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Author : Jacques D. Fleuriot 

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Copyright : 1998 University of Cambridge 

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Conversion to Isar and new proofs by Lawrence C Paulson, 2004 

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GMVT by Benjamin Porter, 2005 

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*) 

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header{* Differentiation *} 

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theory Deriv 

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imports Lim Polynomial 
21164  13 
begin 
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22984  15 
text{*Standard Definitions*} 
21164  16 

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definition 

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deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool" 
21164  19 
{*Differentiation: D is derivative of function f at x*} 
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("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where 
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"DERIV f x :> D = ((%h. (f(x + h)  f x) / h)  0 > D)" 
21164  22 

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consts 

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Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)" 

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primrec 

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"Bolzano_bisect P a b 0 = (a,b)" 

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"Bolzano_bisect P a b (Suc n) = 

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(let (x,y) = Bolzano_bisect P a b n 

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in if P(x, (x+y)/2) then ((x+y)/2, y) 

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else (x, (x+y)/2))" 

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subsection {* Derivatives *} 

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lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h)  f(x))/h)  0 > D)" 
21164  36 
by (simp add: deriv_def) 
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lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h)  f(x))/h)  0 > D" 
21164  39 
by (simp add: deriv_def) 
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lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0" 

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by (simp add: deriv_def) 

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lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1" 
23398  45 
by (simp add: deriv_def cong: LIM_cong) 
21164  46 

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lemma add_diff_add: 

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fixes a b c d :: "'a::ab_group_add" 

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shows "(a + c)  (b + d) = (a  b) + (c  d)" 

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by simp 

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lemma DERIV_add: 

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"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E" 

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by (simp only: deriv_def add_diff_add add_divide_distrib LIM_add) 
21164  55 

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lemma DERIV_minus: 

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"DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x.  f x) x :>  D" 

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by (simp only: deriv_def minus_diff_minus divide_minus_left LIM_minus) 
21164  59 

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lemma DERIV_diff: 

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"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x  g x) x :> D  E" 

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by (simp only: diff_def DERIV_add DERIV_minus) 

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lemma DERIV_add_minus: 

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"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x +  g x) x :> D +  E" 

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by (simp only: DERIV_add DERIV_minus) 

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lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x" 

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proof (unfold isCont_iff) 

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assume "DERIV f x :> D" 

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hence "(\<lambda>h. (f(x+h)  f(x)) / h)  0 > D" 
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by (rule DERIV_D) 
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hence "(\<lambda>h. (f(x+h)  f(x)) / h * h)  0 > D * 0" 
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by (intro LIM_mult LIM_ident) 
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hence "(\<lambda>h. (f(x+h)  f(x)) * (h / h))  0 > 0" 
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by simp 
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hence "(\<lambda>h. f(x+h)  f(x))  0 > 0" 
23398  78 
by (simp cong: LIM_cong) 
21164  79 
thus "(\<lambda>h. f(x+h))  0 > f(x)" 
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by (simp add: LIM_def) 

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qed 

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lemma DERIV_mult_lemma: 

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fixes a b c d :: "'a::real_field" 
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shows "(a * b  c * d) / h = a * ((b  d) / h) + ((a  c) / h) * d" 
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by (simp add: diff_minus add_divide_distrib [symmetric] ring_distribs) 
21164  87 

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lemma DERIV_mult': 

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assumes f: "DERIV f x :> D" 

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assumes g: "DERIV g x :> E" 

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shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x" 

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proof (unfold deriv_def) 

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from f have "isCont f x" 

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by (rule DERIV_isCont) 

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hence "(\<lambda>h. f(x+h))  0 > f x" 

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by (simp only: isCont_iff) 

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hence "(\<lambda>h. f(x+h) * ((g(x+h)  g x) / h) + 
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((f(x+h)  f x) / h) * g x) 
21164  99 
 0 > f x * E + D * g x" 
22613  100 
by (intro LIM_add LIM_mult LIM_const DERIV_D f g) 
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thus "(\<lambda>h. (f(x+h) * g(x+h)  f x * g x) / h) 
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 0 > f x * E + D * g x" 
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by (simp only: DERIV_mult_lemma) 

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qed 

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lemma DERIV_mult: 

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"[ DERIV f x :> Da; DERIV g x :> Db ] 

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==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))" 

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by (drule (1) DERIV_mult', simp only: mult_commute add_commute) 

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lemma DERIV_unique: 

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"[ DERIV f x :> D; DERIV f x :> E ] ==> D = E" 

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apply (simp add: deriv_def) 

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apply (blast intro: LIM_unique) 

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done 

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text{*Differentiation of finite sum*} 

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lemma DERIV_sumr [rule_format (no_asm)]: 

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"(\<forall>r. m \<le> r & r < (m + n) > DERIV (%x. f r x) x :> (f' r x)) 

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> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)" 

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apply (induct "n") 

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apply (auto intro: DERIV_add) 

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done 

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text{*Alternative definition for differentiability*} 

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lemma DERIV_LIM_iff: 

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"((%h. (f(a + h)  f(a)) / h)  0 > D) = 
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((%x. (f(x)f(a)) / (xa))  a > D)" 
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apply (rule iffI) 

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apply (drule_tac k=" a" in LIM_offset) 

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apply (simp add: diff_minus) 

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apply (drule_tac k="a" in LIM_offset) 

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apply (simp add: add_commute) 

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done 

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lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z)  f(x)) / (zx))  x > D)" 
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by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff) 
21164  140 

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lemma inverse_diff_inverse: 

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"\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk> 

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\<Longrightarrow> inverse a  inverse b =  (inverse a * (a  b) * inverse b)" 

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by (simp add: algebra_simps) 
21164  145 

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lemma DERIV_inverse_lemma: 

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"\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk> 
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\<Longrightarrow> (inverse a  inverse b) / h 
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=  (inverse a * ((a  b) / h) * inverse b)" 
21164  150 
by (simp add: inverse_diff_inverse) 
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lemma DERIV_inverse': 

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assumes der: "DERIV f x :> D" 

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assumes neq: "f x \<noteq> 0" 

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shows "DERIV (\<lambda>x. inverse (f x)) x :>  (inverse (f x) * D * inverse (f x))" 

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(is "DERIV _ _ :> ?E") 

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proof (unfold DERIV_iff2) 

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from der have lim_f: "f  x > f x" 

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by (rule DERIV_isCont [unfolded isCont_def]) 

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from neq have "0 < norm (f x)" by simp 

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with LIM_D [OF lim_f] obtain s 

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where s: "0 < s" 

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and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z  x) < s\<rbrakk> 

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\<Longrightarrow> norm (f z  f x) < norm (f x)" 

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by fast 

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show "(\<lambda>z. (inverse (f z)  inverse (f x)) / (z  x))  x > ?E" 
21164  169 
proof (rule LIM_equal2 [OF s]) 
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fix z 
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assume "z \<noteq> x" "norm (z  x) < s" 
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hence "norm (f z  f x) < norm (f x)" by (rule less_fx) 

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hence "f z \<noteq> 0" by auto 

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thus "(inverse (f z)  inverse (f x)) / (z  x) = 
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 (inverse (f z) * ((f z  f x) / (z  x)) * inverse (f x))" 
21164  176 
using neq by (rule DERIV_inverse_lemma) 
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next 

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from der have "(\<lambda>z. (f z  f x) / (z  x))  x > D" 
21164  179 
by (unfold DERIV_iff2) 
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thus "(\<lambda>z.  (inverse (f z) * ((f z  f x) / (z  x)) * inverse (f x))) 
21164  181 
 x > ?E" 
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by (intro LIM_mult LIM_inverse LIM_minus LIM_const lim_f neq) 
21164  183 
qed 
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qed 

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lemma DERIV_divide: 

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"\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk> 
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\<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x  f x * E) / (g x * g x)" 
21164  189 
apply (subgoal_tac "f x *  (inverse (g x) * E * inverse (g x)) + 
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D * inverse (g x) = (D * g x  f x * E) / (g x * g x)") 

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apply (erule subst) 

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apply (unfold divide_inverse) 

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apply (erule DERIV_mult') 

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apply (erule (1) DERIV_inverse') 

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apply (simp add: ring_distribs nonzero_inverse_mult_distrib) 
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apply (simp add: mult_ac) 
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done 

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lemma DERIV_power_Suc: 

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fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}" 
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assumes f: "DERIV f x :> D" 
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shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)" 
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proof (induct n) 
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case 0 

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show ?case by (simp add: power_Suc f) 

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case (Suc k) 

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from DERIV_mult' [OF f Suc] show ?case 

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apply (simp only: of_nat_Suc ring_distribs mult_1_left) 
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apply (simp only: power_Suc algebra_simps) 
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done 
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qed 

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lemma DERIV_power: 

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fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}" 
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assumes f: "DERIV f x :> D" 
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shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n  Suc 0))" 
21164  217 
by (cases "n", simp, simp add: DERIV_power_Suc f) 
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29975  220 
text {* Caratheodory formulation of derivative at a point *} 
21164  221 

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lemma CARAT_DERIV: 

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"(DERIV f x :> l) = 

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(\<exists>g. (\<forall>z. f z  f x = g z * (zx)) & isCont g x & g x = l)" 
21164  225 
(is "?lhs = ?rhs") 
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proof 

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assume der: "DERIV f x :> l" 

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show "\<exists>g. (\<forall>z. f z  f x = g z * (zx)) \<and> isCont g x \<and> g x = l" 
21164  229 
proof (intro exI conjI) 
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let ?g = "(%z. if z = x then l else (f z  f x) / (zx))" 
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show "\<forall>z. f z  f x = ?g z * (zx)" by simp 
21164  232 
show "isCont ?g x" using der 
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by (simp add: isCont_iff DERIV_iff diff_minus 

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cong: LIM_equal [rule_format]) 

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show "?g x = l" by simp 

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qed 

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next 

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assume "?rhs" 

239 
then obtain g where 

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"(\<forall>z. f z  f x = g z * (zx))" and "isCont g x" and "g x = l" by blast 
21164  241 
thus "(DERIV f x :> l)" 
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by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong) 
21164  243 
qed 
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lemma DERIV_chain': 

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assumes f: "DERIV f x :> D" 

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assumes g: "DERIV g (f x) :> E" 

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shows "DERIV (\<lambda>x. g (f x)) x :> E * D" 
21164  249 
proof (unfold DERIV_iff2) 
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obtain d where d: "\<forall>y. g y  g (f x) = d y * (y  f x)" 
21164  251 
and cont_d: "isCont d (f x)" and dfx: "d (f x) = E" 
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using CARAT_DERIV [THEN iffD1, OF g] by fast 

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from f have "f  x > f x" 

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by (rule DERIV_isCont [unfolded isCont_def]) 

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with cont_d have "(\<lambda>z. d (f z))  x > d (f x)" 

21239  256 
by (rule isCont_LIM_compose) 
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hence "(\<lambda>z. d (f z) * ((f z  f x) / (z  x))) 
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 x > d (f x) * D" 
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by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]]) 
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thus "(\<lambda>z. (g (f z)  g (f x)) / (z  x))  x > E * D" 
21164  261 
by (simp add: d dfx real_scaleR_def) 
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qed 

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(* let's do the standard proof though theorem *) 

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(* LIM_mult2 follows from a NS proof *) 

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lemma DERIV_cmult: 

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"DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D" 

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by (drule DERIV_mult' [OF DERIV_const], simp) 

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(* standard version *) 

272 
lemma DERIV_chain: "[ DERIV f (g x) :> Da; DERIV g x :> Db ] ==> DERIV (f o g) x :> Da * Db" 

273 
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute) 

274 

275 
lemma DERIV_chain2: "[ DERIV f (g x) :> Da; DERIV g x :> Db ] ==> DERIV (%x. f (g x)) x :> Da * Db" 

276 
by (auto dest: DERIV_chain simp add: o_def) 

277 

278 
(*derivative of linear multiplication*) 

279 
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c" 

23069
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280 
by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp) 
21164  281 

282 
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n  Suc 0))" 

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283 
apply (cut_tac DERIV_power [OF DERIV_ident]) 
21164  284 
apply (simp add: real_scaleR_def real_of_nat_def) 
285 
done 

286 

287 
text{*Power of 1*} 

288 

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289 
lemma DERIV_inverse: 
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290 
fixes x :: "'a::{real_normed_field,recpower}" 
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291 
shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> ((inverse x ^ Suc (Suc 0)))" 
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292 
by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc) 
21164  293 

294 
text{*Derivative of inverse*} 

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295 
lemma DERIV_inverse_fun: 
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296 
fixes x :: "'a::{real_normed_field,recpower}" 
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297 
shows "[ DERIV f x :> d; f(x) \<noteq> 0 ] 
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298 
==> DERIV (%x. inverse(f x)) x :> ( (d * inverse(f(x) ^ Suc (Suc 0))))" 
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299 
by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib) 
21164  300 

301 
text{*Derivative of quotient*} 

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302 
lemma DERIV_quotient: 
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303 
fixes x :: "'a::{real_normed_field,recpower}" 
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304 
shows "[ DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 ] 
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305 
==> DERIV (%y. f(y) / (g y)) x :> (d*g(x)  (e*f(x))) / (g(x) ^ Suc (Suc 0))" 
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306 
by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc) 
21164  307 

29975  308 
lemma lemma_DERIV_subst: "[ DERIV f x :> D; D = E ] ==> DERIV f x :> E" 
309 
by auto 

310 

22984  311 

312 
subsection {* Differentiability predicate *} 

21164  313 

29169  314 
definition 
315 
differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool" 

316 
(infixl "differentiable" 60) where 

317 
"f differentiable x = (\<exists>D. DERIV f x :> D)" 

318 

319 
lemma differentiableE [elim?]: 

320 
assumes "f differentiable x" 

321 
obtains df where "DERIV f x :> df" 

322 
using prems unfolding differentiable_def .. 

323 

21164  324 
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D" 
325 
by (simp add: differentiable_def) 

326 

327 
lemma differentiableI: "DERIV f x :> D ==> f differentiable x" 

328 
by (force simp add: differentiable_def) 

329 

29169  330 
lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x" 
331 
by (rule DERIV_ident [THEN differentiableI]) 

332 

333 
lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x" 

334 
by (rule DERIV_const [THEN differentiableI]) 

21164  335 

29169  336 
lemma differentiable_compose: 
337 
assumes f: "f differentiable (g x)" 

338 
assumes g: "g differentiable x" 

339 
shows "(\<lambda>x. f (g x)) differentiable x" 

340 
proof  

341 
from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" .. 

342 
moreover 

343 
from `g differentiable x` obtain dg where "DERIV g x :> dg" .. 

344 
ultimately 

345 
have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2) 

346 
thus ?thesis by (rule differentiableI) 

347 
qed 

348 

349 
lemma differentiable_sum [simp]: 

21164  350 
assumes "f differentiable x" 
351 
and "g differentiable x" 

352 
shows "(\<lambda>x. f x + g x) differentiable x" 

353 
proof  

29169  354 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 
355 
moreover 

356 
from `g differentiable x` obtain dg where "DERIV g x :> dg" .. 

357 
ultimately 

358 
have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add) 

359 
thus ?thesis by (rule differentiableI) 

360 
qed 

361 

362 
lemma differentiable_minus [simp]: 

363 
assumes "f differentiable x" 

364 
shows "(\<lambda>x.  f x) differentiable x" 

365 
proof  

366 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 

367 
hence "DERIV (\<lambda>x.  f x) x :>  df" by (rule DERIV_minus) 

368 
thus ?thesis by (rule differentiableI) 

21164  369 
qed 
370 

29169  371 
lemma differentiable_diff [simp]: 
21164  372 
assumes "f differentiable x" 
29169  373 
assumes "g differentiable x" 
21164  374 
shows "(\<lambda>x. f x  g x) differentiable x" 
29169  375 
unfolding diff_minus using prems by simp 
376 

377 
lemma differentiable_mult [simp]: 

378 
assumes "f differentiable x" 

379 
assumes "g differentiable x" 

380 
shows "(\<lambda>x. f x * g x) differentiable x" 

21164  381 
proof  
29169  382 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 
21164  383 
moreover 
29169  384 
from `g differentiable x` obtain dg where "DERIV g x :> dg" .. 
385 
ultimately 

386 
have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult) 

387 
thus ?thesis by (rule differentiableI) 

21164  388 
qed 
389 

29169  390 
lemma differentiable_inverse [simp]: 
391 
assumes "f differentiable x" and "f x \<noteq> 0" 

392 
shows "(\<lambda>x. inverse (f x)) differentiable x" 

21164  393 
proof  
29169  394 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 
395 
hence "DERIV (\<lambda>x. inverse (f x)) x :>  (inverse (f x) * df * inverse (f x))" 

396 
using `f x \<noteq> 0` by (rule DERIV_inverse') 

397 
thus ?thesis by (rule differentiableI) 

21164  398 
qed 
399 

29169  400 
lemma differentiable_divide [simp]: 
401 
assumes "f differentiable x" 

402 
assumes "g differentiable x" and "g x \<noteq> 0" 

403 
shows "(\<lambda>x. f x / g x) differentiable x" 

404 
unfolding divide_inverse using prems by simp 

405 

406 
lemma differentiable_power [simp]: 

407 
fixes f :: "'a::{recpower,real_normed_field} \<Rightarrow> 'a" 

408 
assumes "f differentiable x" 

409 
shows "(\<lambda>x. f x ^ n) differentiable x" 

410 
by (induct n, simp, simp add: power_Suc prems) 

411 

22984  412 

21164  413 
subsection {* Nested Intervals and Bisection *} 
414 

415 
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison). 

416 
All considerably tidied by lcp.*} 

417 

418 
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)" 

419 
apply (induct "no") 

420 
apply (auto intro: order_trans) 

421 
done 

422 

423 
lemma f_inc_g_dec_Beq_f: "[ \<forall>n. f(n) \<le> f(Suc n); 

424 
\<forall>n. g(Suc n) \<le> g(n); 

425 
\<forall>n. f(n) \<le> g(n) ] 

426 
==> Bseq (f :: nat \<Rightarrow> real)" 

427 
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI) 

428 
apply (induct_tac "n") 

429 
apply (auto intro: order_trans) 

430 
apply (rule_tac y = "g (Suc na)" in order_trans) 

431 
apply (induct_tac [2] "na") 

432 
apply (auto intro: order_trans) 

433 
done 

434 

435 
lemma f_inc_g_dec_Beq_g: "[ \<forall>n. f(n) \<le> f(Suc n); 

436 
\<forall>n. g(Suc n) \<le> g(n); 

437 
\<forall>n. f(n) \<le> g(n) ] 

438 
==> Bseq (g :: nat \<Rightarrow> real)" 

439 
apply (subst Bseq_minus_iff [symmetric]) 

440 
apply (rule_tac g = "%x.  (f x)" in f_inc_g_dec_Beq_f) 

441 
apply auto 

442 
done 

443 

444 
lemma f_inc_imp_le_lim: 

445 
fixes f :: "nat \<Rightarrow> real" 

446 
shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f" 

447 
apply (rule linorder_not_less [THEN iffD1]) 

448 
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc) 

449 
apply (drule real_less_sum_gt_zero) 

450 
apply (drule_tac x = "f n +  lim f" in spec, safe) 

451 
apply (drule_tac P = "%na. no\<le>na > ?Q na" and x = "no + n" in spec, auto) 

452 
apply (subgoal_tac "lim f \<le> f (no + n) ") 

453 
apply (drule_tac no=no and m=n in lemma_f_mono_add) 

454 
apply (auto simp add: add_commute) 

455 
apply (induct_tac "no") 

456 
apply simp 

457 
apply (auto intro: order_trans simp add: diff_minus abs_if) 

458 
done 

459 

460 
lemma lim_uminus: "convergent g ==> lim (%x.  g x) =  (lim g)" 

461 
apply (rule LIMSEQ_minus [THEN limI]) 

462 
apply (simp add: convergent_LIMSEQ_iff) 

463 
done 

464 

465 
lemma g_dec_imp_lim_le: 

466 
fixes g :: "nat \<Rightarrow> real" 

467 
shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n" 

468 
apply (subgoal_tac " (g n) \<le>  (lim g) ") 

469 
apply (cut_tac [2] f = "%x.  (g x)" in f_inc_imp_le_lim) 

470 
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric]) 

471 
done 

472 

473 
lemma lemma_nest: "[ \<forall>n. f(n) \<le> f(Suc n); 

474 
\<forall>n. g(Suc n) \<le> g(n); 

475 
\<forall>n. f(n) \<le> g(n) ] 

476 
==> \<exists>l m :: real. l \<le> m & ((\<forall>n. f(n) \<le> l) & f > l) & 

477 
((\<forall>n. m \<le> g(n)) & g > m)" 

478 
apply (subgoal_tac "monoseq f & monoseq g") 

479 
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc) 

480 
apply (subgoal_tac "Bseq f & Bseq g") 

481 
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g) 

482 
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff) 

483 
apply (rule_tac x = "lim f" in exI) 

484 
apply (rule_tac x = "lim g" in exI) 

485 
apply (auto intro: LIMSEQ_le) 

486 
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff) 

487 
done 

488 

489 
lemma lemma_nest_unique: "[ \<forall>n. f(n) \<le> f(Suc n); 

490 
\<forall>n. g(Suc n) \<le> g(n); 

491 
\<forall>n. f(n) \<le> g(n); 

492 
(%n. f(n)  g(n)) > 0 ] 

493 
==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f > l) & 

494 
((\<forall>n. l \<le> g(n)) & g > l)" 

495 
apply (drule lemma_nest, auto) 

496 
apply (subgoal_tac "l = m") 

497 
apply (drule_tac [2] X = f in LIMSEQ_diff) 

498 
apply (auto intro: LIMSEQ_unique) 

499 
done 

500 

501 
text{*The universal quantifiers below are required for the declaration 

502 
of @{text Bolzano_nest_unique} below.*} 

503 

504 
lemma Bolzano_bisect_le: 

505 
"a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)" 

506 
apply (rule allI) 

507 
apply (induct_tac "n") 

508 
apply (auto simp add: Let_def split_def) 

509 
done 

510 

511 
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==> 

512 
\<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))" 

513 
apply (rule allI) 

514 
apply (induct_tac "n") 

515 
apply (auto simp add: Bolzano_bisect_le Let_def split_def) 

516 
done 

517 

518 
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==> 

519 
\<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)" 

520 
apply (rule allI) 

521 
apply (induct_tac "n") 

522 
apply (auto simp add: Bolzano_bisect_le Let_def split_def) 

523 
done 

524 

525 
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)" 

526 
apply (auto) 

527 
apply (drule_tac f = "%u. (1/2) *u" in arg_cong) 

528 
apply (simp) 

529 
done 

530 

531 
lemma Bolzano_bisect_diff: 

532 
"a \<le> b ==> 

533 
snd(Bolzano_bisect P a b n)  fst(Bolzano_bisect P a b n) = 

534 
(ba) / (2 ^ n)" 

535 
apply (induct "n") 

536 
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def) 

537 
done 

538 

539 
lemmas Bolzano_nest_unique = 

540 
lemma_nest_unique 

541 
[OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le] 

542 

543 

544 
lemma not_P_Bolzano_bisect: 

545 
assumes P: "!!a b c. [ P(a,b); P(b,c); a \<le> b; b \<le> c] ==> P(a,c)" 

546 
and notP: "~ P(a,b)" 

547 
and le: "a \<le> b" 

548 
shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))" 

549 
proof (induct n) 

23441  550 
case 0 show ?case using notP by simp 
21164  551 
next 
552 
case (Suc n) 

553 
thus ?case 

554 
by (auto simp del: surjective_pairing [symmetric] 

555 
simp add: Let_def split_def Bolzano_bisect_le [OF le] 

556 
P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"]) 

557 
qed 

558 

559 
(*Now we repackage P_prem as a formula*) 

560 
lemma not_P_Bolzano_bisect': 

561 
"[ \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c > P(a,c); 

562 
~ P(a,b); a \<le> b ] ==> 

563 
\<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))" 

564 
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE]) 

565 

566 

567 

568 
lemma lemma_BOLZANO: 

569 
"[ \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c > P(a,c); 

570 
\<forall>x. \<exists>d::real. 0 < d & 

571 
(\<forall>a b. a \<le> x & x \<le> b & (ba) < d > P(a,b)); 

572 
a \<le> b ] 

573 
==> P(a,b)" 

574 
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+) 

575 
apply (rule LIMSEQ_minus_cancel) 

576 
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero) 

577 
apply (rule ccontr) 

578 
apply (drule not_P_Bolzano_bisect', assumption+) 

579 
apply (rename_tac "l") 

580 
apply (drule_tac x = l in spec, clarify) 

581 
apply (simp add: LIMSEQ_def) 

582 
apply (drule_tac P = "%r. 0<r > ?Q r" and x = "d/2" in spec) 

583 
apply (drule_tac P = "%r. 0<r > ?Q r" and x = "d/2" in spec) 

584 
apply (drule real_less_half_sum, auto) 

585 
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec) 

586 
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec) 

587 
apply safe 

588 
apply (simp_all (no_asm_simp)) 

589 
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) 

590 
apply (simp (no_asm_simp) add: abs_if) 

591 
apply (rule real_sum_of_halves [THEN subst]) 

592 
apply (rule add_strict_mono) 

593 
apply (simp_all add: diff_minus [symmetric]) 

594 
done 

595 

596 

597 
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) > P(a,c)) & 

598 
(\<forall>x. \<exists>d::real. 0 < d & 

599 
(\<forall>a b. a \<le> x & x \<le> b & (ba) < d > P(a,b)))) 

600 
> (\<forall>a b. a \<le> b > P(a,b))" 

601 
apply clarify 

602 
apply (blast intro: lemma_BOLZANO) 

603 
done 

604 

605 

606 
subsection {* Intermediate Value Theorem *} 

607 

608 
text {*Prove Contrapositive by Bisection*} 

609 

610 
lemma IVT: "[ f(a::real) \<le> (y::real); y \<le> f(b); 

611 
a \<le> b; 

612 
(\<forall>x. a \<le> x & x \<le> b > isCont f x) ] 

613 
==> \<exists>x. a \<le> x & x \<le> b & f(x) = y" 

614 
apply (rule contrapos_pp, assumption) 

615 
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) 

616 
apply safe 

617 
apply simp_all 

618 
apply (simp add: isCont_iff LIM_def) 

619 
apply (rule ccontr) 

620 
apply (subgoal_tac "a \<le> x & x \<le> b") 

621 
prefer 2 

622 
apply simp 

623 
apply (drule_tac P = "%d. 0<d > ?P d" and x = 1 in spec, arith) 

624 
apply (drule_tac x = x in spec)+ 

625 
apply simp 

626 
apply (drule_tac P = "%r. ?P r > (\<exists>s>0. ?Q r s) " and x = "\<bar>y  f x\<bar>" in spec) 

627 
apply safe 

628 
apply simp 

629 
apply (drule_tac x = s in spec, clarify) 

630 
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe) 

631 
apply (drule_tac x = "bax" in spec) 

632 
apply (simp_all add: abs_if) 

633 
apply (drule_tac x = "aax" in spec) 

634 
apply (case_tac "x \<le> aa", simp_all) 

635 
done 

636 

637 
lemma IVT2: "[ f(b::real) \<le> (y::real); y \<le> f(a); 

638 
a \<le> b; 

639 
(\<forall>x. a \<le> x & x \<le> b > isCont f x) 

640 
] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y" 

641 
apply (subgoal_tac " f a \<le> y & y \<le>  f b", clarify) 

642 
apply (drule IVT [where f = "%x.  f x"], assumption) 

643 
apply (auto intro: isCont_minus) 

644 
done 

645 

646 
(*HOL style here: objectlevel formulations*) 

647 
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b & 

648 
(\<forall>x. a \<le> x & x \<le> b > isCont f x)) 

649 
> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)" 

650 
apply (blast intro: IVT) 

651 
done 

652 

653 
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b & 

654 
(\<forall>x. a \<le> x & x \<le> b > isCont f x)) 

655 
> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)" 

656 
apply (blast intro: IVT2) 

657 
done 

658 

29975  659 

660 
subsection {* Boundedness of continuous functions *} 

661 

21164  662 
text{*By bisection, function continuous on closed interval is bounded above*} 
663 

664 
lemma isCont_bounded: 

665 
"[ a \<le> b; \<forall>x. a \<le> x & x \<le> b > isCont f x ] 

666 
==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b > f(x) \<le> M" 

667 
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) 

668 
apply safe 

669 
apply simp_all 

670 
apply (rename_tac x xa ya M Ma) 

671 
apply (cut_tac x = M and y = Ma in linorder_linear, safe) 

672 
apply (rule_tac x = Ma in exI, clarify) 

673 
apply (cut_tac x = xb and y = xa in linorder_linear, force) 

674 
apply (rule_tac x = M in exI, clarify) 

675 
apply (cut_tac x = xb and y = xa in linorder_linear, force) 

676 
apply (case_tac "a \<le> x & x \<le> b") 

677 
apply (rule_tac [2] x = 1 in exI) 

678 
prefer 2 apply force 

679 
apply (simp add: LIM_def isCont_iff) 

680 
apply (drule_tac x = x in spec, auto) 

681 
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl) 

682 
apply (drule_tac x = 1 in spec, auto) 

683 
apply (rule_tac x = s in exI, clarify) 

684 
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify) 

685 
apply (drule_tac x = "xax" in spec) 

686 
apply (auto simp add: abs_ge_self) 

687 
done 

688 

689 
text{*Refine the above to existence of least upper bound*} 

690 

691 
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) > 

692 
(\<exists>t. isLub UNIV S t)" 

693 
by (blast intro: reals_complete) 

694 

695 
lemma isCont_has_Ub: "[ a \<le> b; \<forall>x. a \<le> x & x \<le> b > isCont f x ] 

696 
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b > f(x) \<le> M) & 

697 
(\<forall>N. N < M > (\<exists>x. a \<le> x & x \<le> b & N < f(x)))" 

698 
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)" 

699 
in lemma_reals_complete) 

700 
apply auto 

701 
apply (drule isCont_bounded, assumption) 

702 
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def) 

703 
apply (rule exI, auto) 

704 
apply (auto dest!: spec simp add: linorder_not_less) 

705 
done 

706 

707 
text{*Now show that it attains its upper bound*} 

708 

709 
lemma isCont_eq_Ub: 

710 
assumes le: "a \<le> b" 

711 
and con: "\<forall>x::real. a \<le> x & x \<le> b > isCont f x" 

712 
shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b > f(x) \<le> M) & 

713 
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)" 

714 
proof  

715 
from isCont_has_Ub [OF le con] 

716 
obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" 

717 
and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x" by blast 

718 
show ?thesis 

719 
proof (intro exI, intro conjI) 

720 
show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1) 

721 
show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M" 

722 
proof (rule ccontr) 

723 
assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)" 

724 
with M1 have M3: "\<forall>x. a \<le> x & x \<le> b > f x < M" 

725 
by (fastsimp simp add: linorder_not_le [symmetric]) 

726 
hence "\<forall>x. a \<le> x & x \<le> b > isCont (%x. inverse (M  f x)) x" 

727 
by (auto simp add: isCont_inverse isCont_diff con) 

728 
from isCont_bounded [OF le this] 

729 
obtain k where k: "!!x. a \<le> x & x \<le> b > inverse (M  f x) \<le> k" by auto 

730 
have Minv: "!!x. a \<le> x & x \<le> b > 0 < inverse (M  f (x))" 

29667  731 
by (simp add: M3 algebra_simps) 
21164  732 
have "!!x. a \<le> x & x \<le> b > inverse (M  f x) < k+1" using k 
733 
by (auto intro: order_le_less_trans [of _ k]) 

734 
with Minv 

735 
have "!!x. a \<le> x & x \<le> b > inverse(k+1) < inverse(inverse(M  f x))" 

736 
by (intro strip less_imp_inverse_less, simp_all) 

737 
hence invlt: "!!x. a \<le> x & x \<le> b > inverse(k+1) < M  f x" 

738 
by simp 

739 
have "M  inverse (k+1) < M" using k [of a] Minv [of a] le 

740 
by (simp, arith) 

741 
from M2 [OF this] 

742 
obtain x where ax: "a \<le> x & x \<le> b & M  inverse(k+1) < f x" .. 

743 
thus False using invlt [of x] by force 

744 
qed 

745 
qed 

746 
qed 

747 

748 

749 
text{*Same theorem for lower bound*} 

750 

751 
lemma isCont_eq_Lb: "[ a \<le> b; \<forall>x. a \<le> x & x \<le> b > isCont f x ] 

752 
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b > M \<le> f(x)) & 

753 
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)" 

754 
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b > isCont (%x.  (f x)) x") 

755 
prefer 2 apply (blast intro: isCont_minus) 

756 
apply (drule_tac f = "(%x.  (f x))" in isCont_eq_Ub) 

757 
apply safe 

758 
apply auto 

759 
done 

760 

761 

762 
text{*Another version.*} 

763 

764 
lemma isCont_Lb_Ub: "[a \<le> b; \<forall>x. a \<le> x & x \<le> b > isCont f x ] 

765 
==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b > L \<le> f(x) & f(x) \<le> M) & 

766 
(\<forall>y. L \<le> y & y \<le> M > (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))" 

767 
apply (frule isCont_eq_Lb) 

768 
apply (frule_tac [2] isCont_eq_Ub) 

769 
apply (assumption+, safe) 

770 
apply (rule_tac x = "f x" in exI) 

771 
apply (rule_tac x = "f xa" in exI, simp, safe) 

772 
apply (cut_tac x = x and y = xa in linorder_linear, safe) 

773 
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl) 

774 
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe) 

775 
apply (rule_tac [2] x = xb in exI) 

776 
apply (rule_tac [4] x = xb in exI, simp_all) 

777 
done 

778 

779 

29975  780 
subsection {* Local extrema *} 
781 

21164  782 
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*} 
783 

784 
lemma DERIV_left_inc: 

785 
fixes f :: "real => real" 

786 
assumes der: "DERIV f x :> l" 

787 
and l: "0 < l" 

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) 

23441  799 
show "0<s" using s . 
21164  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_left_dec: 

815 
fixes f :: "real => real" 

816 
assumes der: "DERIV f x :> l" 

817 
and l: "l < 0" 

818 
shows "\<exists>d > 0. \<forall>h > 0. h < d > f(x) < f(xh)" 

819 
proof  

820 
from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]] 

821 
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)" 

822 
by (simp add: diff_minus) 

823 
then obtain s 

824 
where s: "0 < s" 

825 
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z)  f x) / z  l\<bar> < l" 

826 
by auto 

827 
thus ?thesis 

828 
proof (intro exI conjI strip) 

23441  829 
show "0<s" using s . 
21164  830 
fix h::real 
831 
assume "0 < h" "h < s" 

832 
with all [of "h"] show "f x < f (xh)" 

833 
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric] 

834 
split add: split_if_asm) 

835 
assume "  ((f (xh)  f x) / h) < l" and h: "0 < h" 

836 
with l 

837 
have "0 < (f (xh)  f x) / h" by arith 

838 
thus "f x < f (xh)" 

839 
by (simp add: pos_less_divide_eq h) 

840 
qed 

841 
qed 

842 
qed 

843 

844 
lemma DERIV_local_max: 

845 
fixes f :: "real => real" 

846 
assumes der: "DERIV f x :> l" 

847 
and d: "0 < d" 

848 
and le: "\<forall>y. \<bar>xy\<bar> < d > f(y) \<le> f(x)" 

849 
shows "l = 0" 

850 
proof (cases rule: linorder_cases [of l 0]) 

23441  851 
case equal thus ?thesis . 
21164  852 
next 
853 
case less 

854 
from DERIV_left_dec [OF der less] 

855 
obtain d' where d': "0 < d'" 

856 
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (xh)" by blast 

857 
from real_lbound_gt_zero [OF d d'] 

858 
obtain e where "0 < e \<and> e < d \<and> e < d'" .. 

859 
with lt le [THEN spec [where x="xe"]] 

860 
show ?thesis by (auto simp add: abs_if) 

861 
next 

862 
case greater 

863 
from DERIV_left_inc [OF der greater] 

864 
obtain d' where d': "0 < d'" 

865 
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast 

866 
from real_lbound_gt_zero [OF d d'] 

867 
obtain e where "0 < e \<and> e < d \<and> e < d'" .. 

868 
with lt le [THEN spec [where x="x+e"]] 

869 
show ?thesis by (auto simp add: abs_if) 

870 
qed 

871 

872 

873 
text{*Similar theorem for a local minimum*} 

874 
lemma DERIV_local_min: 

875 
fixes f :: "real => real" 

876 
shows "[ DERIV f x :> l; 0 < d; \<forall>y. \<bar>xy\<bar> < d > f(x) \<le> f(y) ] ==> l = 0" 

877 
by (drule DERIV_minus [THEN DERIV_local_max], auto) 

878 

879 

880 
text{*In particular, if a function is locally flat*} 

881 
lemma DERIV_local_const: 

882 
fixes f :: "real => real" 

883 
shows "[ DERIV f x :> l; 0 < d; \<forall>y. \<bar>xy\<bar> < d > f(x) = f(y) ] ==> l = 0" 

884 
by (auto dest!: DERIV_local_max) 

885 

29975  886 

887 
subsection {* Rolle's Theorem *} 

888 

21164  889 
text{*Lemma about introducing open ball in open interval*} 
890 
lemma lemma_interval_lt: 

891 
"[ a < x; x < b ] 

892 
==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>xy\<bar> < d > a < y & y < b)" 

27668  893 

22998  894 
apply (simp add: abs_less_iff) 
21164  895 
apply (insert linorder_linear [of "xa" "bx"], safe) 
896 
apply (rule_tac x = "xa" in exI) 

897 
apply (rule_tac [2] x = "bx" in exI, auto) 

898 
done 

899 

900 
lemma lemma_interval: "[ a < x; x < b ] ==> 

901 
\<exists>d::real. 0 < d & (\<forall>y. \<bar>xy\<bar> < d > a \<le> y & y \<le> b)" 

902 
apply (drule lemma_interval_lt, auto) 

903 
apply (auto intro!: exI) 

904 
done 

905 

906 
text{*Rolle's Theorem. 

907 
If @{term f} is defined and continuous on the closed interval 

908 
@{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"}, 

909 
and @{term "f(a) = f(b)"}, 

910 
then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*} 

911 
theorem Rolle: 

912 
assumes lt: "a < b" 

913 
and eq: "f(a) = f(b)" 

914 
and con: "\<forall>x. a \<le> x & x \<le> b > isCont f x" 

915 
and dif [rule_format]: "\<forall>x. a < x & x < b > f differentiable x" 

21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

916 
shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0" 
21164  917 
proof  
918 
have le: "a \<le> b" using lt by simp 

919 
from isCont_eq_Ub [OF le con] 

920 
obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x" 

921 
and alex: "a \<le> x" and xleb: "x \<le> b" 

922 
by blast 

923 
from isCont_eq_Lb [OF le con] 

924 
obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z" 

925 
and alex': "a \<le> x'" and x'leb: "x' \<le> b" 

926 
by blast 

927 
show ?thesis 

928 
proof cases 

929 
assume axb: "a < x & x < b" 

930 
{*@{term f} attains its maximum within the interval*} 

27668  931 
hence ax: "a<x" and xb: "x<b" by arith + 
21164  932 
from lemma_interval [OF ax xb] 
933 
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>xy\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" 

934 
by blast 

935 
hence bound': "\<forall>y. \<bar>xy\<bar> < d \<longrightarrow> f y \<le> f x" using x_max 

936 
by blast 

937 
from differentiableD [OF dif [OF axb]] 

938 
obtain l where der: "DERIV f x :> l" .. 

939 
have "l=0" by (rule DERIV_local_max [OF der d bound']) 

940 
{*the derivative at a local maximum is zero*} 

941 
thus ?thesis using ax xb der by auto 

942 
next 

943 
assume notaxb: "~ (a < x & x < b)" 

944 
hence xeqab: "x=a  x=b" using alex xleb by arith 

945 
hence fb_eq_fx: "f b = f x" by (auto simp add: eq) 

946 
show ?thesis 

947 
proof cases 

948 
assume ax'b: "a < x' & x' < b" 

949 
{*@{term f} attains its minimum within the interval*} 

27668  950 
hence ax': "a<x'" and x'b: "x'<b" by arith+ 
21164  951 
from lemma_interval [OF ax' x'b] 
952 
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" 

953 
by blast 

954 
hence bound': "\<forall>y. \<bar>x'y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min 

955 
by blast 

956 
from differentiableD [OF dif [OF ax'b]] 

957 
obtain l where der: "DERIV f x' :> l" .. 

958 
have "l=0" by (rule DERIV_local_min [OF der d bound']) 

959 
{*the derivative at a local minimum is zero*} 

960 
thus ?thesis using ax' x'b der by auto 

961 
next 

962 
assume notax'b: "~ (a < x' & x' < b)" 

963 
{*@{term f} is constant througout the interval*} 

964 
hence x'eqab: "x'=a  x'=b" using alex' x'leb by arith 

965 
hence fb_eq_fx': "f b = f x'" by (auto simp add: eq) 

966 
from dense [OF lt] 

967 
obtain r where ar: "a < r" and rb: "r < b" by blast 

968 
from lemma_interval [OF ar rb] 

969 
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>ry\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" 

970 
by blast 

971 
have eq_fb: "\<forall>z. a \<le> z > z \<le> b > f z = f b" 

972 
proof (clarify) 

973 
fix z::real 

974 
assume az: "a \<le> z" and zb: "z \<le> b" 

975 
show "f z = f b" 

976 
proof (rule order_antisym) 

977 
show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb) 

978 
show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb) 

979 
qed 

980 
qed 

981 
have bound': "\<forall>y. \<bar>ry\<bar> < d \<longrightarrow> f r = f y" 

982 
proof (intro strip) 

983 
fix y::real 

984 
assume lt: "\<bar>ry\<bar> < d" 

985 
hence "f y = f b" by (simp add: eq_fb bound) 

986 
thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le) 

987 
qed 

988 
from differentiableD [OF dif [OF conjI [OF ar rb]]] 

989 
obtain l where der: "DERIV f r :> l" .. 

990 
have "l=0" by (rule DERIV_local_const [OF der d bound']) 

991 
{*the derivative of a constant function is zero*} 

992 
thus ?thesis using ar rb der by auto 

993 
qed 

994 
qed 

995 
qed 

996 

997 

998 
subsection{*Mean Value Theorem*} 

999 

1000 
lemma lemma_MVT: 

1001 
"f a  (f b  f a)/(ba) * a = f b  (f b  f a)/(ba) * (b::real)" 

1002 
proof cases 

1003 
assume "a=b" thus ?thesis by simp 

1004 
next 

1005 
assume "a\<noteq>b" 

1006 
hence ba: "ba \<noteq> 0" by arith 

1007 
show ?thesis 

1008 
by (rule real_mult_left_cancel [OF ba, THEN iffD1], 

1009 
simp add: right_diff_distrib, 

1010 
simp add: left_diff_distrib) 

1011 
qed 

1012 

1013 
theorem MVT: 

1014 
assumes lt: "a < b" 

1015 
and con: "\<forall>x. a \<le> x & x \<le> b > isCont f x" 

1016 
and dif [rule_format]: "\<forall>x. a < x & x < b > f differentiable x" 

21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1017 
shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l & 
21164  1018 
(f(b)  f(a) = (ba) * l)" 
1019 
proof  

1020 
let ?F = "%x. f x  ((f b  f a) / (ba)) * x" 

1021 
have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con 

23069
cdfff0241c12
rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents:
23044
diff
changeset

1022 
by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident) 
21164  1023 
have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x" 
1024 
proof (clarify) 

1025 
fix x::real 

1026 
assume ax: "a < x" and xb: "x < b" 

1027 
from differentiableD [OF dif [OF conjI [OF ax xb]]] 

1028 
obtain l where der: "DERIV f x :> l" .. 

1029 
show "?F differentiable x" 

1030 
by (rule differentiableI [where D = "l  (f b  f a)/(ba)"], 

1031 
blast intro: DERIV_diff DERIV_cmult_Id der) 

1032 
qed 

1033 
from Rolle [where f = ?F, OF lt lemma_MVT contF difF] 

1034 
obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0" 

1035 
by blast 

1036 
have "DERIV (%x. ((f b  f a)/(ba)) * x) z :> (f b  f a)/(ba)" 

1037 
by (rule DERIV_cmult_Id) 

1038 
hence derF: "DERIV (\<lambda>x. ?F x + (f b  f a) / (b  a) * x) z 

1039 
:> 0 + (f b  f a) / (b  a)" 

1040 
by (rule DERIV_add [OF der]) 

1041 
show ?thesis 

1042 
proof (intro exI conjI) 

23441  1043 
show "a < z" using az . 
1044 
show "z < b" using zb . 

21164  1045 
show "f b  f a = (b  a) * ((f b  f a)/(ba))" by (simp) 
1046 
show "DERIV f z :> ((f b  f a)/(ba))" using derF by simp 

1047 
qed 

1048 
qed 

1049 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1050 
lemma MVT2: 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1051 
"[ a < b; \<forall>x. a \<le> x & x \<le> b > DERIV f x :> f'(x) ] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1052 
==> \<exists>z::real. a < z & z < b & (f b  f a = (b  a) * f'(z))" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1053 
apply (drule MVT) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1054 
apply (blast intro: DERIV_isCont) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1055 
apply (force dest: order_less_imp_le simp add: differentiable_def) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1056 
apply (blast dest: DERIV_unique order_less_imp_le) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1057 
done 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1058 

21164  1059 

1060 
text{*A function is constant if its derivative is 0 over an interval.*} 

1061 

1062 
lemma DERIV_isconst_end: 

1063 
fixes f :: "real => real" 

1064 
shows "[ a < b; 

1065 
\<forall>x. a \<le> x & x \<le> b > isCont f x; 

1066 
\<forall>x. a < x & x < b > DERIV f x :> 0 ] 

1067 
==> f b = f a" 

1068 
apply (drule MVT, assumption) 

1069 
apply (blast intro: differentiableI) 

1070 
apply (auto dest!: DERIV_unique simp add: diff_eq_eq) 

1071 
done 

1072 

1073 
lemma DERIV_isconst1: 

1074 
fixes f :: "real => real" 

1075 
shows "[ a < b; 

1076 
\<forall>x. a \<le> x & x \<le> b > isCont f x; 

1077 
\<forall>x. a < x & x < b > DERIV f x :> 0 ] 

1078 
==> \<forall>x. a \<le> x & x \<le> b > f x = f a" 

1079 
apply safe 

1080 
apply (drule_tac x = a in order_le_imp_less_or_eq, safe) 

1081 
apply (drule_tac b = x in DERIV_isconst_end, auto) 

1082 
done 

1083 

1084 
lemma DERIV_isconst2: 

1085 
fixes f :: "real => real" 

1086 
shows "[ a < b; 

1087 
\<forall>x. a \<le> x & x \<le> b > isCont f x; 

1088 
\<forall>x. a < x & x < b > DERIV f x :> 0; 

1089 
a \<le> x; x \<le> b ] 

1090 
==> f x = f a" 

1091 
apply (blast dest: DERIV_isconst1) 

1092 
done 

1093 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1094 
lemma DERIV_isconst3: fixes a b x y :: real 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1095 
assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1096 
assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1097 
shows "f x = f y" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1098 
proof (cases "x = y") 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1099 
case False 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1100 
let ?a = "min x y" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1101 
let ?b = "max x y" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1102 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1103 
have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1104 
proof (rule allI, rule impI) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1105 
fix z :: real assume "?a \<le> z \<and> z \<le> ?b" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1106 
hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1107 
hence "z \<in> {a<..<b}" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1108 
thus "DERIV f z :> 0" by (rule derivable) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1109 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1110 
hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1111 
and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1112 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1113 
have "?a < ?b" using `x \<noteq> y` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1114 
from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1115 
show ?thesis by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1116 
qed auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

1117 

21164  1118 
lemma DERIV_isconst_all: 
1119 
fixes f :: "real => real" 

1120 
shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)" 

1121 
apply (rule linorder_cases [of x y]) 

1122 
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+ 

1123 
done 

1124 

1125 
lemma DERIV_const_ratio_const: 

21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1126 
fixes f :: "real => real" 
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1127 
shows "[a \<noteq> b; \<forall>x. DERIV f x :> k ] ==> (f(b)  f(a)) = (ba) * k" 
21164  1128 
apply (rule linorder_cases [of a b], auto) 
1129 
apply (drule_tac [!] f = f in MVT) 

1130 
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def) 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23441
diff
changeset

1131 
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus) 
21164  1132 
done 
1133 

1134 
lemma DERIV_const_ratio_const2: 

21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1135 
fixes f :: "real => real" 
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1136 
shows "[a \<noteq> b; \<forall>x. DERIV f x :> k ] ==> (f(b)  f(a))/(ba) = k" 
21164  1137 
apply (rule_tac c1 = "ba" in real_mult_right_cancel [THEN iffD1]) 
1138 
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc) 

1139 
done 

1140 

1141 
lemma real_average_minus_first [simp]: "((a + b) /2  a) = (ba)/(2::real)" 

1142 
by (simp) 

1143 

1144 
lemma real_average_minus_second [simp]: "((b + a)/2  a) = (ba)/(2::real)" 

1145 
by (simp) 

1146 

1147 
text{*Gallileo's "trick": average velocity = av. of end velocities*} 

1148 

1149 
lemma DERIV_const_average: 

1150 
fixes v :: "real => real" 

1151 
assumes neq: "a \<noteq> (b::real)" 

1152 
and der: "\<forall>x. DERIV v x :> k" 

1153 
shows "v ((a + b)/2) = (v a + v b)/2" 

1154 
proof (cases rule: linorder_cases [of a b]) 

1155 
case equal with neq show ?thesis by simp 

1156 
next 

1157 
case less 

1158 
have "(v b  v a) / (b  a) = k" 

1159 
by (rule DERIV_const_ratio_const2 [OF neq der]) 

1160 
hence "(ba) * ((v b  v a) / (ba)) = (ba) * k" by simp 

1161 
moreover have "(v ((a + b) / 2)  v a) / ((a + b) / 2  a) = k" 

1162 
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq) 

1163 
ultimately show ?thesis using neq by force 

1164 
next 

1165 
case greater 

1166 
have "(v b  v a) / (b  a) = k" 

1167 
by (rule DERIV_const_ratio_const2 [OF neq der]) 

1168 
hence "(ba) * ((v b  v a) / (ba)) = (ba) * k" by simp 

1169 
moreover have " (v ((b + a) / 2)  v a) / ((b + a) / 2  a) = k" 

1170 
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq) 

1171 
ultimately show ?thesis using neq by (force simp add: add_commute) 

1172 
qed 

1173 

1174 

29975  1175 
subsection {* Continuous injective functions *} 
1176 

21164  1177 
text{*Dull lemma: an continuous injection on an interval must have a 
1178 
strict maximum at an end point, not in the middle.*} 

1179 

1180 
lemma lemma_isCont_inj: 

1181 
fixes f :: "real \<Rightarrow> real" 

1182 
assumes d: "0 < d" 

1183 
and inj [rule_format]: "\<forall>z. \<bar>zx\<bar> \<le> d > g(f z) = z" 

1184 
and cont: "\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z" 

1185 
shows "\<exists>z. \<bar>zx\<bar> \<le> d & f x < f z" 

1186 
proof (rule ccontr) 

1187 
assume "~ (\<exists>z. \<bar>zx\<bar> \<le> d & f x < f z)" 

1188 
hence all [rule_format]: "\<forall>z. \<bar>z  x\<bar> \<le> d > f z \<le> f x" by auto 

1189 
show False 

1190 
proof (cases rule: linorder_le_cases [of "f(xd)" "f(x+d)"]) 

1191 
case le 

1192 
from d cont all [of "x+d"] 

1193 
have flef: "f(x+d) \<le> f x" 

1194 
and xlex: "x  d \<le> x" 

1195 
and cont': "\<forall>z. x  d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z" 

1196 
by (auto simp add: abs_if) 

1197 
from IVT [OF le flef xlex cont'] 

1198 
obtain x' where "xd \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast 

1199 
moreover 

1200 
hence "g(f x') = g (f(x+d))" by simp 

1201 
ultimately show False using d inj [of x'] inj [of "x+d"] 

22998  1202 
by (simp add: abs_le_iff) 
21164  1203 
next 
1204 
case ge 

1205 
from d cont all [of "xd"] 

1206 
have flef: "f(xd) \<le> f x" 

1207 
and xlex: "x \<le> x+d" 

1208 
and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" 

1209 
by (auto simp add: abs_if) 

1210 
from IVT2 [OF ge flef xlex cont'] 

1211 
obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(xd)" by blast 

1212 
moreover 

1213 
hence "g(f x') = g (f(xd))" by simp 

1214 
ultimately show False using d inj [of x'] inj [of "xd"] 

22998  1215 
by (simp add: abs_le_iff) 
21164  1216 
qed 
1217 
qed 

1218 

1219 

1220 
text{*Similar version for lower bound.*} 

1221 

1222 
lemma lemma_isCont_inj2: 

1223 
fixes f g :: "real \<Rightarrow> real" 

1224 
shows "[0 < d; \<forall>z. \<bar>zx\<bar> \<le> d > g(f z) = z; 

1225 
\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z ] 

1226 
==> \<exists>z. \<bar>zx\<bar> \<le> d & f z < f x" 

1227 
apply (insert lemma_isCont_inj 

1228 
[where f = "%x.  f x" and g = "%y. g(y)" and x = x and d = d]) 

1229 
apply (simp add: isCont_minus linorder_not_le) 

1230 
done 

1231 

1232 
text{*Show there's an interval surrounding @{term "f(x)"} in 

1233 
@{text "f[[x  d, x + d]]"} .*} 

1234 

1235 
lemma isCont_inj_range: 

1236 
fixes f :: "real \<Rightarrow> real" 

1237 
assumes d: "0 < d" 

1238 
and inj: "\<forall>z. \<bar>zx\<bar> \<le> d > g(f z) = z" 

1239 
and cont: "\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z" 

1240 
shows "\<exists>e>0. \<forall>y. \<bar>y  f x\<bar> \<le> e > (\<exists>z. \<bar>zx\<bar> \<le> d & f z = y)" 

1241 
proof  

1242 
have "xd \<le> x+d" "\<forall>z. xd \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d 

22998  1243 
by (auto simp add: abs_le_iff) 
21164  1244 
from isCont_Lb_Ub [OF this] 
1245 
obtain L M 

1246 
where all1 [rule_format]: "\<forall>z. xd \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M" 

1247 
and all2 [rule_format]: 

1248 
"\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. xd \<le> z \<and> z \<le> x+d \<and> f z = y)" 

1249 
by auto 

1250 
with d have "L \<le> f x & f x \<le> M" by simp 

1251 
moreover have "L \<noteq> f x" 

1252 
proof  

1253 
from lemma_isCont_inj2 [OF d inj cont] 

1254 
obtain u where "\<bar>u  x\<bar> \<le> d" "f u < f x" by auto 

1255 
thus ?thesis using all1 [of u] by arith 

1256 
qed 

1257 
moreover have "f x \<noteq> M" 

1258 
proof  

1259 
from lemma_isCont_inj [OF d inj cont] 

1260 
obtain u where "\<bar>u  x\<bar> \<le> d" "f x < f u" by auto 

1261 
thus ?thesis using all1 [of u] by arith 

1262 
qed 

1263 
ultimately have "L < f x & f x < M" by arith 

1264 
hence "0 < f x  L" "0 < M  f x" by arith+ 

1265 
from real_lbound_gt_zero [OF this] 

1266 
obtain e where e: "0 < e" "e < f x  L" "e < M  f x" by auto 

1267 
thus ?thesis 

1268 
proof (intro exI conjI) 

23441  1269 
show "0<e" using e(1) . 
21164  1270 
show "\<forall>y. \<bar>y  f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z  x\<bar> \<le> d \<and> f z = y)" 
1271 
proof (intro strip) 

1272 
fix y::real 

1273 
assume "\<bar>y  f x\<bar> \<le> e" 

1274 
with e have "L \<le> y \<and> y \<le> M" by arith 

1275 
from all2 [OF this] 

1276 
obtain z where "x  d \<le> z" "z \<le> x + d" "f z = y" by blast 

27668  1277 
thus "\<exists>z. \<bar>z  x\<bar> \<le> d \<and> f z = y" 
22998  1278 
by (force simp add: abs_le_iff) 
21164  1279 
qed 
1280 
qed 

1281 
qed 

1282 

1283 

1284 
text{*Continuity of inverse function*} 

1285 

1286 
lemma isCont_inverse_function: 

1287 
fixes f g :: "real \<Rightarrow> real" 

1288 
assumes d: "0 < d" 

1289 
and inj: "\<forall>z. \<bar>zx\<bar> \<le> d > g(f z) = z" 

1290 
and cont: "\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z" 

1291 
shows "isCont g (f x)" 

1292 
proof (simp add: isCont_iff LIM_eq) 

1293 
show "\<forall>r. 0 < r \<longrightarrow> 

1294 
(\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z)  g(f x)\<bar> < r)" 

1295 
proof (intro strip) 

1296 
fix r::real 

1297 
assume r: "0<r" 

1298 
from real_lbound_gt_zero [OF r d] 

1299 
obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast 

1300 
with inj cont 

1301 
have e_simps: "\<forall>z. \<bar>zx\<bar> \<le> e > g (f z) = z" 

1302 
"\<forall>z. \<bar>zx\<bar> \<le> e > isCont f z" by auto 

1303 
from isCont_inj_range [OF e this] 

1304 
obtain e' where e': "0 < e'" 

1305 
and all: "\<forall>y. \<bar>y  f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z  x\<bar> \<le> e \<and> f z = y)" 

1306 
by blast 

1307 
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" 

1308 
proof (intro exI conjI) 

23441  1309 
show "0<e'" using e' . 
21164  1310 
show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z)  g (f x)\<bar> < r" 
1311 
proof (intro strip) 

1312 
fix z::real 

1313 
assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'" 

1314 
with e e_lt e_simps all [rule_format, of "f x + z"] 

1315 
show "\<bar>g (f x + z)  g (f x)\<bar> < r" by force 

1316 
qed 

1317 
qed 

1318 
qed 

1319 
qed 

1320 

23041  1321 
text {* Derivative of inverse function *} 
1322 

1323 
lemma DERIV_inverse_function: 

1324 
fixes f g :: "real \<Rightarrow> real" 

1325 
assumes der: "DERIV f (g x) :> D" 

1326 
assumes neq: "D \<noteq> 0" 

23044  1327 
assumes a: "a < x" and b: "x < b" 
1328 
assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y" 

23041  1329 
assumes cont: "isCont g x" 
1330 
shows "DERIV g x :> inverse D" 

1331 
unfolding DERIV_iff2 

23044  1332 
proof (rule LIM_equal2) 
1333 
show "0 < min (x  a) (b  x)" 

27668  1334 
using a b by arith 
23044  1335 
next 
23041  1336 
fix y 
23044  1337 
assume "norm (y  x) < min (x  a) (b  x)" 
27668  1338 
hence "a < y" and "y < b" 
23044  1339 
by (simp_all add: abs_less_iff) 
23041  1340 
thus "(g y  g x) / (y  x) = 
1341 
inverse ((f (g y)  x) / (g y  g x))" 

1342 
by (simp add: inj) 

1343 
next 

1344 
have "(\<lambda>z. (f z  f (g x)) / (z  g x))  g x > D" 

1345 
by (rule der [unfolded DERIV_iff2]) 

1346 
hence 1: "(\<lambda>z. (f z  x) / (z  g x))  g x > D" 

23044  1347 
using inj a b by simp 
23041  1348 
have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y  x) < d \<longrightarrow> g y \<noteq> g x" 
1349 
proof (safe intro!: exI) 

23044  1350 
show "0 < min (x  a) (b  x)" 
1351 
using a b by simp 

23041  1352 
next 
1353 
fix y 

23044  1354 
assume "norm (y  x) < min (x  a) (b  x)" 
1355 
hence y: "a < y" "y < b" 

1356 
by (simp_all add: abs_less_iff) 

23041  1357 
assume "g y = g x" 
1358 
hence "f (g y) = f (g x)" by simp 

23044  1359 
hence "y = x" using inj y a b by simp 
23041  1360 
also assume "y \<noteq> x" 
1361 
finally show False by simp 

1362 
qed 

1363 
have "(\<lambda>y. (f (g y)  x) / (g y  g x))  x > D" 

1364 
using cont 1 2 by (rule isCont_LIM_compose2) 

1365 
thus "(\<lambda>y. inverse ((f (g y)  x) / (g y  g x))) 

1366 
 x > inverse D" 

1367 
using neq by (rule LIM_inverse) 

1368 
qed 

1369 

29975  1370 

1371 
subsection {* Generalized Mean Value Theorem *} 

1372 

21164  1373 
theorem GMVT: 
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1374 
fixes a b :: real 
21164  1375 
assumes alb: "a < b" 
1376 
and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x" 

1377 
and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x" 

1378 
and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x" 

1379 
and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x" 

1380 
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)" 

1381 
proof  

1382 
let ?h = "\<lambda>x. (f b  f a)*(g x)  (g b  g a)*(f x)" 

1383 
from prems have "a < b" by simp 

1384 
moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x" 

1385 
proof  

1386 
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b  f a) x" by simp 

1387 
with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b  f a) * g x) x" 

1388 
by (auto intro: isCont_mult) 

1389 
moreover 

1390 
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b  g a) x" by simp 

1391 
with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b  g a) * f x) x" 

1392 
by (auto intro: isCont_mult) 

1393 
ultimately show ?thesis 

1394 
by (fastsimp intro: isCont_diff) 

1395 
qed 

1396 
moreover 

1397 
have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x" 

1398 
proof  

1399 
have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b  f a) differentiable x" by (simp add: differentiable_const) 

1400 
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) 

1401 
moreover 

1402 
have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b  g a) differentiable x" by (simp add: differentiable_const) 

1403 
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) 

1404 
ultimately show ?thesis by (simp add: differentiable_diff) 

1405 
qed 

1406 
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) 

1407 
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" .. 

1408 
then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b  ?h a = (b  a) * l" .. 

1409 

1410 
from cdef have cint: "a < c \<and> c < b" by auto 

1411 
with gd have "g differentiable c" by simp 
