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permissions  rwrr 
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(* 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" 
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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: ring_simps) 
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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 
21164  171 
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))" 
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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" 
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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 right_distrib mult_ac add_ac) 
21164  210 
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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(*  *) 

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(* Caratheodory formulation of derivative at a point: standard proof *) 

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(*  *) 

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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  227 
(is "?lhs = ?rhs") 
228 
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  231 
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  234 
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" 

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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  243 
thus "(DERIV f x :> l)" 
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by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong) 
21164  245 
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  251 
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  253 
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  258 
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  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" 

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282 
by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp) 
21164  283 

284 
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n  Suc 0))" 

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285 
apply (cut_tac DERIV_power [OF DERIV_ident]) 
21164  286 
apply (simp add: real_scaleR_def real_of_nat_def) 
287 
done 

288 

289 
text{*Power of 1*} 

290 

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291 
lemma DERIV_inverse: 
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292 
fixes x :: "'a::{real_normed_field,recpower}" 
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293 
shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> ((inverse x ^ Suc (Suc 0)))" 
23069
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294 
by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc) 
21164  295 

296 
text{*Derivative of inverse*} 

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297 
lemma DERIV_inverse_fun: 
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298 
fixes x :: "'a::{real_normed_field,recpower}" 
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299 
shows "[ DERIV f x :> d; f(x) \<noteq> 0 ] 
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300 
==> DERIV (%x. inverse(f x)) x :> ( (d * inverse(f(x) ^ Suc (Suc 0))))" 
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301 
by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib) 
21164  302 

303 
text{*Derivative of quotient*} 

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304 
lemma DERIV_quotient: 
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305 
fixes x :: "'a::{real_normed_field,recpower}" 
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306 
shows "[ DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 ] 
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307 
==> DERIV (%y. f(y) / (g y)) x :> (d*g(x)  (e*f(x))) / (g(x) ^ Suc (Suc 0))" 
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308 
by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc) 
21164  309 

22984  310 

311 
subsection {* Differentiability predicate *} 

21164  312 

29169  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 

21164  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 

29169  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]) 

21164  334 

29169  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]: 

21164  349 
assumes "f differentiable x" 
350 
and "g differentiable x" 

351 
shows "(\<lambda>x. f x + g x) differentiable x" 

352 
proof  

29169  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) 

21164  368 
qed 
369 

29169  370 
lemma differentiable_diff [simp]: 
21164  371 
assumes "f differentiable x" 
29169  372 
assumes "g differentiable x" 
21164  373 
shows "(\<lambda>x. f x  g x) differentiable x" 
29169  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" 

21164  380 
proof  
29169  381 
from `f differentiable x` obtain df where "DERIV f x :> df" .. 
21164  382 
moreover 
29169  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) 

21164  387 
qed 
388 

29169  389 
lemma differentiable_inverse [simp]: 
390 
assumes "f differentiable x" and "f x \<noteq> 0" 

391 
shows "(\<lambda>x. inverse (f x)) differentiable x" 

21164  392 
proof  
29169  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) 

21164  397 
qed 
398 

29169  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 

22984  411 

21164  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 
(ba) / (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) 

23441  549 
case 0 show ?case using notP by simp 
21164  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 repackage 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 & (ba) < 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 & (ba) < 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 = "bax" in spec) 

631 
apply (simp_all add: abs_if) 

632 
apply (drule_tac x = "aax" 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: objectlevel 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 = "xax" 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) 

23441  793 
show "0<s" using s . 
21164  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(xh)" 

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) 

23441  823 
show "0<s" using s . 
21164  824 
fix h::real 
825 
assume "0 < h" "h < s" 

826 
with all [of "h"] show "f x < f (xh)" 

827 
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric] 

828 
split add: split_if_asm) 

829 
assume "  ((f (xh)  f x) / h) < l" and h: "0 < h" 

830 
with l 

831 
have "0 < (f (xh)  f x) / h" by arith 

832 
thus "f x < f (xh)" 

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>xy\<bar> < d > f(y) \<le> f(x)" 

843 
shows "l = 0" 

844 
proof (cases rule: linorder_cases [of l 0]) 

23441  845 
case equal thus ?thesis . 
21164  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 (xh)" 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="xe"]] 

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>xy\<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>xy\<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>xy\<bar> < d > a < y & y < b)" 

27668  884 

22998  885 
apply (simp add: abs_less_iff) 
21164  886 
apply (insert linorder_linear [of "xa" "bx"], safe) 
887 
apply (rule_tac x = "xa" in exI) 

888 
apply (rule_tac [2] x = "bx" in exI, auto) 

889 
done 

890 

891 
lemma lemma_interval: "[ a < x; x < b ] ==> 

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

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

907 
shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0" 
21164  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*} 

27668  922 
hence ax: "a<x" and xb: "x<b" by arith + 
21164  923 
from lemma_interval [OF ax xb] 
924 
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>xy\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" 

925 
by blast 

926 
hence bound': "\<forall>y. \<bar>xy\<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*} 

27668  941 
hence ax': "a<x'" and x'b: "x'<b" by arith+ 
21164  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>ry\<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>ry\<bar> < d \<longrightarrow> f r = f y" 

973 
proof (intro strip) 

974 
fix y::real 

975 
assume lt: "\<bar>ry\<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)/(ba) * a = f b  (f b  f a)/(ba) * (b::real)" 

993 
proof cases 

994 
assume "a=b" thus ?thesis by simp 

995 
next 

996 
assume "a\<noteq>b" 

997 
hence ba: "ba \<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" 

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

1008 
shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l & 
21164  1009 
(f(b)  f(a) = (ba) * l)" 
1010 
proof  

1011 
let ?F = "%x. f x  ((f b  f a) / (ba)) * x" 

1012 
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

1013 
by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident) 
21164  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)/(ba)"], 

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)/(ba)) * x) z :> (f b  f a)/(ba)" 

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) 

23441  1034 
show "a < z" using az . 
1035 
show "z < b" using zb . 

21164  1036 
show "f b  f a = (b  a) * ((f b  f a)/(ba))" by (simp) 
1037 
show "DERIV f z :> ((f b  f a)/(ba))" 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: 

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

1084 
fixes f :: "real => real" 
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1085 
shows "[a \<noteq> b; \<forall>x. DERIV f x :> k ] ==> (f(b)  f(a)) = (ba) * k" 
21164  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) 

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

1089 
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus) 
21164  1090 
done 
1091 

1092 
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

1093 
fixes f :: "real => real" 
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1094 
shows "[a \<noteq> b; \<forall>x. DERIV f x :> k ] ==> (f(b)  f(a))/(ba) = k" 
21164  1095 
apply (rule_tac c1 = "ba" 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) = (ba)/(2::real)" 

1100 
by (simp) 

1101 

1102 
lemma real_average_minus_second [simp]: "((b + a)/2  a) = (ba)/(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 "(ba) * ((v b  v a) / (ba)) = (ba) * 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 "(ba) * ((v b  v a) / (ba)) = (ba) * 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>zx\<bar> \<le> d > g(f z) = z" 

1140 
and cont: "\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z" 

1141 
shows "\<exists>z. \<bar>zx\<bar> \<le> d & f x < f z" 

1142 
proof (rule ccontr) 

1143 
assume "~ (\<exists>z. \<bar>zx\<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(xd)" "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 "xd \<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"] 

22998  1158 
by (simp add: abs_le_iff) 
21164  1159 
next 
1160 
case ge 

1161 
from d cont all [of "xd"] 

1162 
have flef: "f(xd) \<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(xd)" by blast 

1168 
moreover 

1169 
hence "g(f x') = g (f(xd))" by simp 

1170 
ultimately show False using d inj [of x'] inj [of "xd"] 

22998  1171 
by (simp add: abs_le_iff) 
21164  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>zx\<bar> \<le> d > g(f z) = z; 

1181 
\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z ] 

1182 
==> \<exists>z. \<bar>zx\<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>zx\<bar> \<le> d > g(f z) = z" 

1195 
and cont: "\<forall>z. \<bar>zx\<bar> \<le> d > isCont f z" 

1196 
shows "\<exists>e>0. \<forall>y. \<bar>y  f x\<bar> \<le> e > (\<exists>z. \<bar>zx\<bar> \<le> d & f z = y)" 

1197 
proof  

1198 
have "xd \<le> x+d" "\<forall>z. xd \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d 

22998  1199 
by (auto simp add: abs_le_iff) 
21164  1200 
from isCont_Lb_Ub [OF this] 
1201 
obtain L M 

1202 
where all1 [rule_format]: "\<forall>z. xd \<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. xd \<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) 

23441  1225 
show "0<e" using e(1) . 
21164  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 

27668  1233 
thus "\<exists>z. \<bar>z  x\<bar> \<le> d \<and> f z = y" 
22998  1234 
by (force simp add: abs_le_iff) 
21164  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>zx\<bar> \<le> d > g(f z) = z" 

1246 
and cont: "\<forall>z. \<bar>zx\<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>zx\<bar> \<le> e > g (f z) = z" 

1258 
"\<forall>z. \<bar>zx\<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) 

23441  1265 
show "0<e'" using e' . 
21164  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 

23041  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" 

23044  1283 
assumes a: "a < x" and b: "x < b" 
1284 
assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y" 

23041  1285 
assumes cont: "isCont g x" 
1286 
shows "DERIV g x :> inverse D" 

1287 
unfolding DERIV_iff2 

23044  1288 
proof (rule LIM_equal2) 
1289 
show "0 < min (x  a) (b  x)" 

27668  1290 
using a b by arith 
23044  1291 
next 
23041  1292 
fix y 
23044  1293 
assume "norm (y  x) < min (x  a) (b  x)" 
27668  1294 
hence "a < y" and "y < b" 
23044  1295 
by (simp_all add: abs_less_iff) 
23041  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" 

23044  1303 
using inj a b by simp 
23041  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) 

23044  1306 
show "0 < min (x  a) (b  x)" 
1307 
using a b by simp 

23041  1308 
next 
1309 
fix y 

23044  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) 

23041  1313 
assume "g y = g x" 
1314 
hence "f (g y) = f (g x)" by simp 

23044  1315 
hence "y = x" using inj y a b by simp 
23041  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 

21164  1326 
theorem GMVT: 
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset

1327 
fixes a b :: real 
21164  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 
{ 

23441  1376 
have "DERIV (\<lambda>x. (f b  f a) * g x) c :> g'c * (f b  f a)" 
21164  1377 
apply (insert DERIV_const [where k="f b  f a"]) 
1378 
apply (drule meta_spec [of _ c]) 

23441  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)" 

21164  1382 
apply (insert DERIV_const [where k="g b  g a"]) 
1383 
apply (drule meta_spec [of _ c]) 

23441  1384 
apply (drule DERIV_mult [OF _ f'cdef]) 
1385 
by simp 

21164  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))" 

22998  1401 
by (simp add: mult_ac add_ac right_diff_distrib) 
21164  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 

23255  1412 
lemma lemma_DERIV_subst: "[ DERIV f x :> D; D = E ] ==> DERIV f x :> E" 
1413 
by auto 

1414 

29470
1851088a1f87
convert Deriv.thy to use new Polynomial library (incomplete)
huffman
parents:
29169
diff
changeset

1415 

26120
2dd43c63c100
Includes the derivates of polynomials  reals specific content of Poly
chaieb
parents:
23477
diff
changeset

1416 
subsection {* Derivatives of univariate polynomials *} 
2dd43c63c100
Includes the derivates of polynomials  reals specific content of Poly
chaieb
parents:
