author | huffman |
Tue, 12 Dec 2006 07:11:58 +0100 | |
changeset 21785 | 885667874dfc |
parent 21784 | e76faa6e65fd |
child 21810 | b2d23672b003 |
permissions | -rw-r--r-- |
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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 |
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begin |
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text{*Standard and Nonstandard Definitions*} |
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definition |
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deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool" |
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--{*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)" |
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definition |
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nsderiv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool" |
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("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where |
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"NSDERIV f x :> D = (\<forall>h \<in> Infinitesimal - {0}. |
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(( *f* f)(star_of x + h) |
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- star_of (f x))/h @= star_of D)" |
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definition |
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differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool" |
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(infixl "differentiable" 60) where |
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"f differentiable x = (\<exists>D. DERIV f x :> D)" |
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definition |
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NSdifferentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool" |
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(infixl "NSdifferentiable" 60) where |
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"f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)" |
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definition |
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increment :: "[real=>real,real,hypreal] => hypreal" where |
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"increment f x h = (@inc. f NSdifferentiable x & |
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inc = ( *f* f)(hypreal_of_real x + h) - hypreal_of_real (f x))" |
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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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subsubsection {* Purely standard proofs *} |
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lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)" |
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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" |
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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_Id [simp]: "DERIV (\<lambda>x. x) x :> 1" |
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by (simp add: deriv_def divide_self cong: LIM_cong) |
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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) |
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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) |
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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_self) |
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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" |
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by (simp cong: LIM_cong add: divide_self) |
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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_distrib) |
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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) |
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-- 0 --> f x * E + D * g x" |
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by (intro LIM_add LIM_mult2 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)) / (x-a)) -- 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)) / (z-x)) -- x --> D)" |
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by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff) |
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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: right_diff_distrib left_diff_distrib mult_assoc) |
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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)" |
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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" |
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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))" |
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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))) |
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-- x --> ?E" |
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by (intro LIM_mult2 LIM_inverse LIM_minus LIM_const lim_f neq) |
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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)" |
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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: left_diff_distrib 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 :> (of_nat n + 1) * (D * f x ^ n)" |
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proof (induct n) |
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case 0 |
|
230 |
show ?case by (simp add: power_Suc f) |
|
231 |
case (Suc k) |
|
232 |
from DERIV_mult' [OF f Suc] show ?case |
|
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|
233 |
apply (simp only: of_nat_Suc left_distrib mult_1_left) |
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|
234 |
apply (simp only: power_Suc right_distrib mult_ac) |
21164 | 235 |
done |
236 |
qed |
|
237 |
||
238 |
lemma DERIV_power: |
|
21784
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|
239 |
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}" |
21164 | 240 |
assumes f: "DERIV f x :> D" |
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|
241 |
shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))" |
21164 | 242 |
by (cases "n", simp, simp add: DERIV_power_Suc f) |
243 |
||
244 |
||
245 |
(* ------------------------------------------------------------------------ *) |
|
246 |
(* Caratheodory formulation of derivative at a point: standard proof *) |
|
247 |
(* ------------------------------------------------------------------------ *) |
|
248 |
||
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|
249 |
lemma nonzero_mult_divide_cancel_right: |
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|
250 |
"b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)" |
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|
251 |
proof - |
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|
252 |
assume b: "b \<noteq> 0" |
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|
253 |
have "a * b / b = a * (b / b)" by simp |
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|
254 |
also have "\<dots> = a" by (simp add: divide_self b) |
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|
255 |
finally show "a * b / b = a" . |
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|
256 |
qed |
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|
257 |
|
21164 | 258 |
lemma CARAT_DERIV: |
259 |
"(DERIV f x :> l) = |
|
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|
260 |
(\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)" |
21164 | 261 |
(is "?lhs = ?rhs") |
262 |
proof |
|
263 |
assume der: "DERIV f x :> l" |
|
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|
264 |
show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l" |
21164 | 265 |
proof (intro exI conjI) |
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|
266 |
let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))" |
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|
267 |
show "\<forall>z. f z - f x = ?g z * (z-x)" |
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|
268 |
by (simp add: nonzero_mult_divide_cancel_right) |
21164 | 269 |
show "isCont ?g x" using der |
270 |
by (simp add: isCont_iff DERIV_iff diff_minus |
|
271 |
cong: LIM_equal [rule_format]) |
|
272 |
show "?g x = l" by simp |
|
273 |
qed |
|
274 |
next |
|
275 |
assume "?rhs" |
|
276 |
then obtain g where |
|
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|
277 |
"(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast |
21164 | 278 |
thus "(DERIV f x :> l)" |
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|
279 |
by (auto simp add: isCont_iff DERIV_iff nonzero_mult_divide_cancel_right |
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|
280 |
cong: LIM_cong) |
21164 | 281 |
qed |
282 |
||
283 |
lemma DERIV_chain': |
|
284 |
assumes f: "DERIV f x :> D" |
|
285 |
assumes g: "DERIV g (f x) :> E" |
|
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|
286 |
shows "DERIV (\<lambda>x. g (f x)) x :> E * D" |
21164 | 287 |
proof (unfold DERIV_iff2) |
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|
288 |
obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)" |
21164 | 289 |
and cont_d: "isCont d (f x)" and dfx: "d (f x) = E" |
290 |
using CARAT_DERIV [THEN iffD1, OF g] by fast |
|
291 |
from f have "f -- x --> f x" |
|
292 |
by (rule DERIV_isCont [unfolded isCont_def]) |
|
293 |
with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)" |
|
21239 | 294 |
by (rule isCont_LIM_compose) |
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|
295 |
hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x))) |
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|
296 |
-- x --> d (f x) * D" |
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|
297 |
by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]]) |
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|
298 |
thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D" |
21164 | 299 |
by (simp add: d dfx real_scaleR_def) |
300 |
qed |
|
301 |
||
302 |
||
303 |
subsubsection {* Nonstandard proofs *} |
|
304 |
||
305 |
lemma DERIV_NS_iff: |
|
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|
306 |
"(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)" |
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|
307 |
by (simp add: deriv_def LIM_NSLIM_iff) |
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|
308 |
|
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|
309 |
lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --NS> D" |
21164 | 310 |
by (simp add: deriv_def LIM_NSLIM_iff) |
311 |
||
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|
312 |
lemma hnorm_of_hypreal: |
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changeset
|
313 |
"\<And>r. hnorm (( *f* of_real) r::'a::real_normed_div_algebra star) = \<bar>r\<bar>" |
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|
314 |
by transfer (rule norm_of_real) |
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changeset
|
315 |
|
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|
316 |
lemma Infinitesimal_of_hypreal: |
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|
317 |
"x \<in> Infinitesimal \<Longrightarrow> |
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|
318 |
(( *f* of_real) x::'a::real_normed_div_algebra star) \<in> Infinitesimal" |
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|
319 |
apply (rule InfinitesimalI2) |
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|
320 |
apply (drule (1) InfinitesimalD2) |
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huffman
parents:
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changeset
|
321 |
apply (simp add: hnorm_of_hypreal) |
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huffman
parents:
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|
322 |
done |
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huffman
parents:
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changeset
|
323 |
|
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huffman
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changeset
|
324 |
lemma of_hypreal_eq_0_iff: |
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huffman
parents:
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changeset
|
325 |
"\<And>x. (( *f* of_real) x = (0::'a::real_algebra_1 star)) = (x = 0)" |
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huffman
parents:
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changeset
|
326 |
by transfer (rule of_real_eq_0_iff) |
21164 | 327 |
|
328 |
lemma NSDeriv_unique: |
|
329 |
"[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E" |
|
21784
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huffman
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changeset
|
330 |
apply (subgoal_tac "( *f* of_real) epsilon \<in> Infinitesimal - {0::'a star}") |
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huffman
parents:
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changeset
|
331 |
apply (simp only: nsderiv_def) |
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huffman
parents:
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changeset
|
332 |
apply (drule (1) bspec)+ |
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huffman
parents:
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changeset
|
333 |
apply (drule (1) approx_trans3) |
e76faa6e65fd
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huffman
parents:
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changeset
|
334 |
apply simp |
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huffman
parents:
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changeset
|
335 |
apply (simp add: Infinitesimal_of_hypreal Infinitesimal_epsilon) |
e76faa6e65fd
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huffman
parents:
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changeset
|
336 |
apply (simp add: of_hypreal_eq_0_iff hypreal_epsilon_not_zero) |
21164 | 337 |
done |
338 |
||
339 |
text {*First NSDERIV in terms of NSLIM*} |
|
340 |
||
341 |
text{*first equivalence *} |
|
342 |
lemma NSDERIV_NSLIM_iff: |
|
343 |
"(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)" |
|
344 |
apply (simp add: nsderiv_def NSLIM_def, auto) |
|
345 |
apply (drule_tac x = xa in bspec) |
|
346 |
apply (rule_tac [3] ccontr) |
|
347 |
apply (drule_tac [3] x = h in spec) |
|
348 |
apply (auto simp add: mem_infmal_iff starfun_lambda_cancel) |
|
349 |
done |
|
350 |
||
351 |
text{*second equivalence *} |
|
352 |
lemma NSDERIV_NSLIM_iff2: |
|
353 |
"(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)" |
|
354 |
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff diff_minus [symmetric] |
|
355 |
LIM_NSLIM_iff [symmetric]) |
|
356 |
||
357 |
(* while we're at it! *) |
|
358 |
lemma NSDERIV_iff2: |
|
359 |
"(NSDERIV f x :> D) = |
|
360 |
(\<forall>w. |
|
21784
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changeset
|
361 |
w \<noteq> star_of x & w \<approx> star_of x --> |
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huffman
parents:
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changeset
|
362 |
( *f* (%z. (f z - f x) / (z-x))) w \<approx> star_of D)" |
21164 | 363 |
by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def) |
364 |
||
365 |
(*FIXME DELETE*) |
|
21784
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huffman
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changeset
|
366 |
lemma hypreal_not_eq_minus_iff: |
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huffman
parents:
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changeset
|
367 |
"(x \<noteq> a) = (x - a \<noteq> (0::'a::ab_group_add))" |
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huffman
parents:
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|
368 |
by auto |
21164 | 369 |
|
370 |
lemma NSDERIVD5: |
|
371 |
"(NSDERIV f x :> D) ==> |
|
372 |
(\<forall>u. u \<approx> hypreal_of_real x --> |
|
373 |
( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))" |
|
374 |
apply (auto simp add: NSDERIV_iff2) |
|
375 |
apply (case_tac "u = hypreal_of_real x", auto) |
|
376 |
apply (drule_tac x = u in spec, auto) |
|
377 |
apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1) |
|
378 |
apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1]) |
|
379 |
apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ") |
|
380 |
apply (auto simp add: |
|
381 |
approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]] |
|
382 |
Infinitesimal_subset_HFinite [THEN subsetD]) |
|
383 |
done |
|
384 |
||
385 |
lemma NSDERIVD4: |
|
386 |
"(NSDERIV f x :> D) ==> |
|
387 |
(\<forall>h \<in> Infinitesimal. |
|
388 |
(( *f* f)(hypreal_of_real x + h) - |
|
389 |
hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)" |
|
390 |
apply (auto simp add: nsderiv_def) |
|
391 |
apply (case_tac "h = (0::hypreal) ") |
|
392 |
apply (auto simp add: diff_minus) |
|
393 |
apply (drule_tac x = h in bspec) |
|
394 |
apply (drule_tac [2] c = h in approx_mult1) |
|
395 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] |
|
396 |
simp add: diff_minus) |
|
397 |
done |
|
398 |
||
399 |
lemma NSDERIVD3: |
|
400 |
"(NSDERIV f x :> D) ==> |
|
401 |
(\<forall>h \<in> Infinitesimal - {0}. |
|
402 |
(( *f* f)(hypreal_of_real x + h) - |
|
403 |
hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)" |
|
404 |
apply (auto simp add: nsderiv_def) |
|
405 |
apply (rule ccontr, drule_tac x = h in bspec) |
|
406 |
apply (drule_tac [2] c = h in approx_mult1) |
|
407 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] |
|
408 |
simp add: mult_assoc diff_minus) |
|
409 |
done |
|
410 |
||
411 |
text{*Differentiability implies continuity |
|
412 |
nice and simple "algebraic" proof*} |
|
413 |
lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x" |
|
414 |
apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def) |
|
415 |
apply (drule approx_minus_iff [THEN iffD1]) |
|
416 |
apply (drule hypreal_not_eq_minus_iff [THEN iffD1]) |
|
21784
e76faa6e65fd
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huffman
parents:
21404
diff
changeset
|
417 |
apply (drule_tac x = "xa - star_of x" in bspec) |
21164 | 418 |
prefer 2 apply (simp add: add_assoc [symmetric]) |
419 |
apply (auto simp add: mem_infmal_iff [symmetric] add_commute) |
|
21784
e76faa6e65fd
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huffman
parents:
21404
diff
changeset
|
420 |
apply (drule_tac c = "xa - star_of x" in approx_mult1) |
21164 | 421 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] |
21784
e76faa6e65fd
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huffman
parents:
21404
diff
changeset
|
422 |
simp add: mult_assoc nonzero_mult_divide_cancel_right) |
21164 | 423 |
apply (drule_tac x3=D in |
21784
e76faa6e65fd
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huffman
parents:
21404
diff
changeset
|
424 |
HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult, |
21164 | 425 |
THEN mem_infmal_iff [THEN iffD1]]) |
426 |
apply (auto simp add: mult_commute |
|
427 |
intro: approx_trans approx_minus_iff [THEN iffD2]) |
|
428 |
done |
|
429 |
||
430 |
text{*Differentiation rules for combinations of functions |
|
431 |
follow from clear, straightforard, algebraic |
|
432 |
manipulations*} |
|
433 |
text{*Constant function*} |
|
434 |
||
435 |
(* use simple constant nslimit theorem *) |
|
436 |
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)" |
|
437 |
by (simp add: NSDERIV_NSLIM_iff) |
|
438 |
||
439 |
text{*Sum of functions- proved easily*} |
|
440 |
||
441 |
lemma NSDERIV_add: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] |
|
442 |
==> NSDERIV (%x. f x + g x) x :> Da + Db" |
|
443 |
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def) |
|
444 |
apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
445 |
apply (drule_tac b = "star_of Da" and d = "star_of Db" in approx_add) |
21164 | 446 |
apply (auto simp add: diff_def add_ac) |
447 |
done |
|
448 |
||
449 |
text{*Product of functions - Proof is trivial but tedious |
|
450 |
and long due to rearrangement of terms*} |
|
451 |
||
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
452 |
lemma lemma_nsderiv1: |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
453 |
fixes a b c d :: "'a::comm_ring star" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
454 |
shows "(a*b) - (c*d) = (b*(a - c)) + (c*(b - d))" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
455 |
by (simp add: right_diff_distrib mult_ac) |
21164 | 456 |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
457 |
lemma lemma_nsderiv2: |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
458 |
fixes x y z :: "'a::real_normed_field star" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
459 |
shows "[| (x - y) / z = star_of D + yb; z \<noteq> 0; |
21164 | 460 |
z \<in> Infinitesimal; yb \<in> Infinitesimal |] |
461 |
==> x - y \<approx> 0" |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
462 |
apply (simp add: nonzero_divide_eq_eq) |
21164 | 463 |
apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add |
464 |
simp add: mult_assoc mem_infmal_iff [symmetric]) |
|
465 |
apply (erule Infinitesimal_subset_HFinite [THEN subsetD]) |
|
466 |
done |
|
467 |
||
468 |
lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] |
|
469 |
==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))" |
|
470 |
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def) |
|
471 |
apply (auto dest!: spec |
|
472 |
simp add: starfun_lambda_cancel lemma_nsderiv1) |
|
473 |
apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib) |
|
474 |
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+ |
|
475 |
apply (auto simp add: times_divide_eq_right [symmetric] |
|
476 |
simp del: times_divide_eq) |
|
477 |
apply (drule_tac D = Db in lemma_nsderiv2, assumption+) |
|
478 |
apply (drule_tac |
|
479 |
approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]]) |
|
480 |
apply (auto intro!: approx_add_mono1 |
|
481 |
simp add: left_distrib right_distrib mult_commute add_assoc) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
482 |
apply (rule_tac b1 = "star_of Db * star_of (f x)" |
21164 | 483 |
in add_commute [THEN subst]) |
484 |
apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym] |
|
485 |
Infinitesimal_add Infinitesimal_mult |
|
486 |
Infinitesimal_hypreal_of_real_mult |
|
487 |
Infinitesimal_hypreal_of_real_mult2 |
|
488 |
simp add: add_assoc [symmetric]) |
|
489 |
done |
|
490 |
||
491 |
text{*Multiplying by a constant*} |
|
492 |
lemma NSDERIV_cmult: "NSDERIV f x :> D |
|
493 |
==> NSDERIV (%x. c * f x) x :> c*D" |
|
494 |
apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff |
|
495 |
minus_mult_right right_diff_distrib [symmetric]) |
|
496 |
apply (erule NSLIM_const [THEN NSLIM_mult]) |
|
497 |
done |
|
498 |
||
499 |
text{*Negation of function*} |
|
500 |
lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D" |
|
501 |
proof (simp add: NSDERIV_NSLIM_iff) |
|
502 |
assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D" |
|
503 |
hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D" |
|
504 |
by (rule NSLIM_minus) |
|
505 |
have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h" |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
506 |
by (simp add: minus_divide_left diff_def) |
21164 | 507 |
with deriv |
508 |
show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp |
|
509 |
qed |
|
510 |
||
511 |
text{*Subtraction*} |
|
512 |
lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db" |
|
513 |
by (blast dest: NSDERIV_add NSDERIV_minus) |
|
514 |
||
515 |
lemma NSDERIV_diff: |
|
516 |
"[| NSDERIV f x :> Da; NSDERIV g x :> Db |] |
|
517 |
==> NSDERIV (%x. f x - g x) x :> Da-Db" |
|
518 |
apply (simp add: diff_minus) |
|
519 |
apply (blast intro: NSDERIV_add_minus) |
|
520 |
done |
|
521 |
||
522 |
text{* Similarly to the above, the chain rule admits an entirely |
|
523 |
straightforward derivation. Compare this with Harrison's |
|
524 |
HOL proof of the chain rule, which proved to be trickier and |
|
525 |
required an alternative characterisation of differentiability- |
|
526 |
the so-called Carathedory derivative. Our main problem is |
|
527 |
manipulation of terms.*} |
|
528 |
||
529 |
||
530 |
(* lemmas *) |
|
531 |
lemma NSDERIV_zero: |
|
532 |
"[| NSDERIV g x :> D; |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
533 |
( *f* g) (star_of x + xa) = star_of (g x); |
21164 | 534 |
xa \<in> Infinitesimal; |
535 |
xa \<noteq> 0 |
|
536 |
|] ==> D = 0" |
|
537 |
apply (simp add: nsderiv_def) |
|
538 |
apply (drule bspec, auto) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
539 |
apply (rule star_of_approx_iff [THEN iffD1], simp) |
21164 | 540 |
done |
541 |
||
542 |
(* can be proved differently using NSLIM_isCont_iff *) |
|
543 |
lemma NSDERIV_approx: |
|
544 |
"[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |] |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
545 |
==> ( *f* f) (star_of x + h) - star_of (f x) \<approx> 0" |
21164 | 546 |
apply (simp add: nsderiv_def) |
547 |
apply (simp add: mem_infmal_iff [symmetric]) |
|
548 |
apply (rule Infinitesimal_ratio) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
549 |
apply (rule_tac [3] approx_star_of_HFinite, auto) |
21164 | 550 |
done |
551 |
||
552 |
(*--------------------------------------------------------------- |
|
553 |
from one version of differentiability |
|
554 |
||
555 |
f(x) - f(a) |
|
556 |
--------------- \<approx> Db |
|
557 |
x - a |
|
558 |
---------------------------------------------------------------*) |
|
559 |
lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da; |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
560 |
( *f* g) (star_of(x) + xa) \<noteq> star_of (g x); |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
561 |
( *f* g) (star_of(x) + xa) \<approx> star_of (g x) |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
562 |
|] ==> (( *f* f) (( *f* g) (star_of(x) + xa)) |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
563 |
- star_of (f (g x))) |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
564 |
/ (( *f* g) (star_of(x) + xa) - star_of (g x)) |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
565 |
\<approx> star_of(Da)" |
21164 | 566 |
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric]) |
567 |
||
568 |
(*-------------------------------------------------------------- |
|
569 |
from other version of differentiability |
|
570 |
||
571 |
f(x + h) - f(x) |
|
572 |
----------------- \<approx> Db |
|
573 |
h |
|
574 |
--------------------------------------------------------------*) |
|
575 |
lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |] |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
576 |
==> (( *f* g) (star_of(x) + xa) - star_of(g x)) / xa |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
577 |
\<approx> star_of(Db)" |
21164 | 578 |
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel) |
579 |
||
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
580 |
lemma lemma_chain: "(z::'a::real_normed_field star) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
581 |
proof - |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
582 |
assume z: "z \<noteq> 0" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
583 |
have "x * y = x * (inverse z * z) * y" by (simp add: z) |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
584 |
thus ?thesis by (simp add: mult_assoc) |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
585 |
qed |
21164 | 586 |
|
587 |
text{*This proof uses both definitions of differentiability.*} |
|
588 |
lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |] |
|
589 |
==> NSDERIV (f o g) x :> Da * Db" |
|
590 |
apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def |
|
591 |
mem_infmal_iff [symmetric]) |
|
592 |
apply clarify |
|
593 |
apply (frule_tac f = g in NSDERIV_approx) |
|
594 |
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric]) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
595 |
apply (case_tac "( *f* g) (star_of (x) + xa) = star_of (g x) ") |
21164 | 596 |
apply (drule_tac g = g in NSDERIV_zero) |
597 |
apply (auto simp add: divide_inverse) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
598 |
apply (rule_tac z1 = "( *f* g) (star_of (x) + xa) - star_of (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst]) |
21164 | 599 |
apply (erule hypreal_not_eq_minus_iff [THEN iffD1]) |
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
600 |
apply (rule approx_mult_star_of) |
21164 | 601 |
apply (simp_all add: divide_inverse [symmetric]) |
602 |
apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2]) |
|
603 |
apply (blast intro: NSDERIVD2) |
|
604 |
done |
|
605 |
||
606 |
text{*Differentiation of natural number powers*} |
|
607 |
lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1" |
|
608 |
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if) |
|
609 |
||
610 |
lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c" |
|
611 |
by (cut_tac c = c and x = x in NSDERIV_Id [THEN NSDERIV_cmult], simp) |
|
612 |
||
613 |
(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*) |
|
614 |
lemma NSDERIV_inverse: |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
615 |
fixes x :: "'a::{real_normed_field,recpower}" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
616 |
shows "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))" |
21164 | 617 |
apply (simp add: nsderiv_def) |
618 |
apply (rule ballI, simp, clarify) |
|
619 |
apply (frule (1) Infinitesimal_add_not_zero) |
|
620 |
apply (simp add: add_commute) |
|
621 |
(*apply (auto simp add: starfun_inverse_inverse realpow_two |
|
622 |
simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
623 |
apply (simp add: inverse_add nonzero_inverse_mult_distrib [symmetric] power_Suc |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
624 |
nonzero_inverse_minus_eq [symmetric] add_ac mult_ac diff_def |
21164 | 625 |
del: inverse_mult_distrib inverse_minus_eq |
626 |
minus_mult_left [symmetric] minus_mult_right [symmetric]) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
627 |
apply (subst mult_commute, simp add: nonzero_mult_divide_cancel_right) |
21164 | 628 |
apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib |
629 |
del: minus_mult_left [symmetric] minus_mult_right [symmetric]) |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
630 |
apply (rule_tac y = "inverse (- (star_of x * star_of x))" in approx_trans) |
21164 | 631 |
apply (rule inverse_add_Infinitesimal_approx2) |
632 |
apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal |
|
633 |
simp add: inverse_minus_eq [symmetric] HFinite_minus_iff) |
|
634 |
apply (rule Infinitesimal_HFinite_mult2, auto) |
|
635 |
done |
|
636 |
||
637 |
subsubsection {* Equivalence of NS and Standard definitions *} |
|
638 |
||
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
639 |
lemma divideR_eq_divide: "x /# y = x / y" |
21164 | 640 |
by (simp add: real_scaleR_def divide_inverse mult_commute) |
641 |
||
642 |
text{*Now equivalence between NSDERIV and DERIV*} |
|
643 |
lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)" |
|
644 |
by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff) |
|
645 |
||
646 |
(* let's do the standard proof though theorem *) |
|
647 |
(* LIM_mult2 follows from a NS proof *) |
|
648 |
||
649 |
lemma DERIV_cmult: |
|
650 |
"DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D" |
|
651 |
by (drule DERIV_mult' [OF DERIV_const], simp) |
|
652 |
||
653 |
(* standard version *) |
|
654 |
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db" |
|
655 |
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute) |
|
656 |
||
657 |
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db" |
|
658 |
by (auto dest: DERIV_chain simp add: o_def) |
|
659 |
||
660 |
(*derivative of linear multiplication*) |
|
661 |
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c" |
|
662 |
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp) |
|
663 |
||
664 |
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))" |
|
665 |
apply (cut_tac DERIV_power [OF DERIV_Id]) |
|
666 |
apply (simp add: real_scaleR_def real_of_nat_def) |
|
667 |
done |
|
668 |
||
669 |
(* NS version *) |
|
670 |
lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))" |
|
671 |
by (simp add: NSDERIV_DERIV_iff DERIV_pow) |
|
672 |
||
673 |
text{*Power of -1*} |
|
674 |
||
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
675 |
lemma DERIV_inverse: |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
676 |
fixes x :: "'a::{real_normed_field,recpower}" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
677 |
shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
678 |
by (drule DERIV_inverse' [OF DERIV_Id]) (simp add: power_Suc) |
21164 | 679 |
|
680 |
text{*Derivative of inverse*} |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
681 |
lemma DERIV_inverse_fun: |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
682 |
fixes x :: "'a::{real_normed_field,recpower}" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
683 |
shows "[| DERIV f x :> d; f(x) \<noteq> 0 |] |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
684 |
==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
685 |
by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib) |
21164 | 686 |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
687 |
lemma NSDERIV_inverse_fun: |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
688 |
fixes x :: "'a::{real_normed_field,recpower}" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
689 |
shows "[| NSDERIV f x :> d; f(x) \<noteq> 0 |] |
21164 | 690 |
==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))" |
691 |
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc) |
|
692 |
||
693 |
text{*Derivative of quotient*} |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
694 |
lemma DERIV_quotient: |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
695 |
fixes x :: "'a::{real_normed_field,recpower}" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
696 |
shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |] |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
697 |
==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
698 |
by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc) |
21164 | 699 |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
700 |
lemma NSDERIV_quotient: |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
701 |
fixes x :: "'a::{real_normed_field,recpower}" |
21785 | 702 |
shows "[| NSDERIV f x :> d; NSDERIV g x :> e; g(x) \<noteq> 0 |] |
21164 | 703 |
==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x) |
704 |
- (e*f(x))) / (g(x) ^ Suc (Suc 0))" |
|
705 |
by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc) |
|
706 |
||
707 |
lemma CARAT_NSDERIV: "NSDERIV f x :> l ==> |
|
708 |
\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l" |
|
709 |
by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
710 |
mult_commute) |
21164 | 711 |
|
712 |
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))" |
|
713 |
by auto |
|
714 |
||
715 |
lemma CARAT_DERIVD: |
|
716 |
assumes all: "\<forall>z. f z - f x = g z * (z-x)" |
|
717 |
and nsc: "isNSCont g x" |
|
718 |
shows "NSDERIV f x :> g x" |
|
719 |
proof - |
|
720 |
from nsc |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
721 |
have "\<forall>w. w \<noteq> star_of x \<and> w \<approx> star_of x \<longrightarrow> |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
722 |
( *f* g) w * (w - star_of x) / (w - star_of x) \<approx> |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
723 |
star_of (g x)" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
724 |
by (simp add: isNSCont_def nonzero_mult_divide_cancel_right) |
21164 | 725 |
thus ?thesis using all |
726 |
by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong) |
|
727 |
qed |
|
728 |
||
729 |
subsubsection {* Differentiability predicate *} |
|
730 |
||
731 |
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D" |
|
732 |
by (simp add: differentiable_def) |
|
733 |
||
734 |
lemma differentiableI: "DERIV f x :> D ==> f differentiable x" |
|
735 |
by (force simp add: differentiable_def) |
|
736 |
||
737 |
lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D" |
|
738 |
by (simp add: NSdifferentiable_def) |
|
739 |
||
740 |
lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x" |
|
741 |
by (force simp add: NSdifferentiable_def) |
|
742 |
||
743 |
lemma differentiable_const: "(\<lambda>z. a) differentiable x" |
|
744 |
apply (unfold differentiable_def) |
|
745 |
apply (rule_tac x=0 in exI) |
|
746 |
apply simp |
|
747 |
done |
|
748 |
||
749 |
lemma differentiable_sum: |
|
750 |
assumes "f differentiable x" |
|
751 |
and "g differentiable x" |
|
752 |
shows "(\<lambda>x. f x + g x) differentiable x" |
|
753 |
proof - |
|
754 |
from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def) |
|
755 |
then obtain df where "DERIV f x :> df" .. |
|
756 |
moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def) |
|
757 |
then obtain dg where "DERIV g x :> dg" .. |
|
758 |
ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add) |
|
759 |
hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto |
|
760 |
thus ?thesis by (fold differentiable_def) |
|
761 |
qed |
|
762 |
||
763 |
lemma differentiable_diff: |
|
764 |
assumes "f differentiable x" |
|
765 |
and "g differentiable x" |
|
766 |
shows "(\<lambda>x. f x - g x) differentiable x" |
|
767 |
proof - |
|
768 |
from prems have "f differentiable x" by simp |
|
769 |
moreover |
|
770 |
from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def) |
|
771 |
then obtain dg where "DERIV g x :> dg" .. |
|
772 |
then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus) |
|
773 |
hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto |
|
774 |
hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def) |
|
775 |
ultimately |
|
776 |
show ?thesis |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
777 |
by (auto simp: diff_def dest: differentiable_sum) |
21164 | 778 |
qed |
779 |
||
780 |
lemma differentiable_mult: |
|
781 |
assumes "f differentiable x" |
|
782 |
and "g differentiable x" |
|
783 |
shows "(\<lambda>x. f x * g x) differentiable x" |
|
784 |
proof - |
|
785 |
from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def) |
|
786 |
then obtain df where "DERIV f x :> df" .. |
|
787 |
moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def) |
|
788 |
then obtain dg where "DERIV g x :> dg" .. |
|
789 |
ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult) |
|
790 |
hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto |
|
791 |
thus ?thesis by (fold differentiable_def) |
|
792 |
qed |
|
793 |
||
794 |
subsection {*(NS) Increment*} |
|
795 |
lemma incrementI: |
|
796 |
"f NSdifferentiable x ==> |
|
797 |
increment f x h = ( *f* f) (hypreal_of_real(x) + h) - |
|
798 |
hypreal_of_real (f x)" |
|
799 |
by (simp add: increment_def) |
|
800 |
||
801 |
lemma incrementI2: "NSDERIV f x :> D ==> |
|
802 |
increment f x h = ( *f* f) (hypreal_of_real(x) + h) - |
|
803 |
hypreal_of_real (f x)" |
|
804 |
apply (erule NSdifferentiableI [THEN incrementI]) |
|
805 |
done |
|
806 |
||
807 |
(* The Increment theorem -- Keisler p. 65 *) |
|
808 |
lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |] |
|
809 |
==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h" |
|
810 |
apply (frule_tac h = h in incrementI2, simp add: nsderiv_def) |
|
811 |
apply (drule bspec, auto) |
|
812 |
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify) |
|
813 |
apply (frule_tac b1 = "hypreal_of_real (D) + y" |
|
814 |
in hypreal_mult_right_cancel [THEN iffD2]) |
|
815 |
apply (erule_tac [2] V = "(( *f* f) (hypreal_of_real (x) + h) - hypreal_of_real (f x)) / h = hypreal_of_real (D) + y" in thin_rl) |
|
816 |
apply assumption |
|
817 |
apply (simp add: times_divide_eq_right [symmetric]) |
|
818 |
apply (auto simp add: left_distrib) |
|
819 |
done |
|
820 |
||
821 |
lemma increment_thm2: |
|
822 |
"[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |] |
|
823 |
==> \<exists>e \<in> Infinitesimal. increment f x h = |
|
824 |
hypreal_of_real(D)*h + e*h" |
|
825 |
by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm) |
|
826 |
||
827 |
||
828 |
lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |] |
|
829 |
==> increment f x h \<approx> 0" |
|
830 |
apply (drule increment_thm2, |
|
831 |
auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric]) |
|
832 |
apply (erule Infinitesimal_subset_HFinite [THEN subsetD]) |
|
833 |
done |
|
834 |
||
835 |
subsection {* Nested Intervals and Bisection *} |
|
836 |
||
837 |
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison). |
|
838 |
All considerably tidied by lcp.*} |
|
839 |
||
840 |
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)" |
|
841 |
apply (induct "no") |
|
842 |
apply (auto intro: order_trans) |
|
843 |
done |
|
844 |
||
845 |
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n); |
|
846 |
\<forall>n. g(Suc n) \<le> g(n); |
|
847 |
\<forall>n. f(n) \<le> g(n) |] |
|
848 |
==> Bseq (f :: nat \<Rightarrow> real)" |
|
849 |
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI) |
|
850 |
apply (induct_tac "n") |
|
851 |
apply (auto intro: order_trans) |
|
852 |
apply (rule_tac y = "g (Suc na)" in order_trans) |
|
853 |
apply (induct_tac [2] "na") |
|
854 |
apply (auto intro: order_trans) |
|
855 |
done |
|
856 |
||
857 |
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n); |
|
858 |
\<forall>n. g(Suc n) \<le> g(n); |
|
859 |
\<forall>n. f(n) \<le> g(n) |] |
|
860 |
==> Bseq (g :: nat \<Rightarrow> real)" |
|
861 |
apply (subst Bseq_minus_iff [symmetric]) |
|
862 |
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f) |
|
863 |
apply auto |
|
864 |
done |
|
865 |
||
866 |
lemma f_inc_imp_le_lim: |
|
867 |
fixes f :: "nat \<Rightarrow> real" |
|
868 |
shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f" |
|
869 |
apply (rule linorder_not_less [THEN iffD1]) |
|
870 |
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc) |
|
871 |
apply (drule real_less_sum_gt_zero) |
|
872 |
apply (drule_tac x = "f n + - lim f" in spec, safe) |
|
873 |
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto) |
|
874 |
apply (subgoal_tac "lim f \<le> f (no + n) ") |
|
875 |
apply (drule_tac no=no and m=n in lemma_f_mono_add) |
|
876 |
apply (auto simp add: add_commute) |
|
877 |
apply (induct_tac "no") |
|
878 |
apply simp |
|
879 |
apply (auto intro: order_trans simp add: diff_minus abs_if) |
|
880 |
done |
|
881 |
||
882 |
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)" |
|
883 |
apply (rule LIMSEQ_minus [THEN limI]) |
|
884 |
apply (simp add: convergent_LIMSEQ_iff) |
|
885 |
done |
|
886 |
||
887 |
lemma g_dec_imp_lim_le: |
|
888 |
fixes g :: "nat \<Rightarrow> real" |
|
889 |
shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n" |
|
890 |
apply (subgoal_tac "- (g n) \<le> - (lim g) ") |
|
891 |
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim) |
|
892 |
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric]) |
|
893 |
done |
|
894 |
||
895 |
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n); |
|
896 |
\<forall>n. g(Suc n) \<le> g(n); |
|
897 |
\<forall>n. f(n) \<le> g(n) |] |
|
898 |
==> \<exists>l m :: real. l \<le> m & ((\<forall>n. f(n) \<le> l) & f ----> l) & |
|
899 |
((\<forall>n. m \<le> g(n)) & g ----> m)" |
|
900 |
apply (subgoal_tac "monoseq f & monoseq g") |
|
901 |
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc) |
|
902 |
apply (subgoal_tac "Bseq f & Bseq g") |
|
903 |
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g) |
|
904 |
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff) |
|
905 |
apply (rule_tac x = "lim f" in exI) |
|
906 |
apply (rule_tac x = "lim g" in exI) |
|
907 |
apply (auto intro: LIMSEQ_le) |
|
908 |
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff) |
|
909 |
done |
|
910 |
||
911 |
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n); |
|
912 |
\<forall>n. g(Suc n) \<le> g(n); |
|
913 |
\<forall>n. f(n) \<le> g(n); |
|
914 |
(%n. f(n) - g(n)) ----> 0 |] |
|
915 |
==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) & |
|
916 |
((\<forall>n. l \<le> g(n)) & g ----> l)" |
|
917 |
apply (drule lemma_nest, auto) |
|
918 |
apply (subgoal_tac "l = m") |
|
919 |
apply (drule_tac [2] X = f in LIMSEQ_diff) |
|
920 |
apply (auto intro: LIMSEQ_unique) |
|
921 |
done |
|
922 |
||
923 |
text{*The universal quantifiers below are required for the declaration |
|
924 |
of @{text Bolzano_nest_unique} below.*} |
|
925 |
||
926 |
lemma Bolzano_bisect_le: |
|
927 |
"a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)" |
|
928 |
apply (rule allI) |
|
929 |
apply (induct_tac "n") |
|
930 |
apply (auto simp add: Let_def split_def) |
|
931 |
done |
|
932 |
||
933 |
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==> |
|
934 |
\<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))" |
|
935 |
apply (rule allI) |
|
936 |
apply (induct_tac "n") |
|
937 |
apply (auto simp add: Bolzano_bisect_le Let_def split_def) |
|
938 |
done |
|
939 |
||
940 |
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==> |
|
941 |
\<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)" |
|
942 |
apply (rule allI) |
|
943 |
apply (induct_tac "n") |
|
944 |
apply (auto simp add: Bolzano_bisect_le Let_def split_def) |
|
945 |
done |
|
946 |
||
947 |
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)" |
|
948 |
apply (auto) |
|
949 |
apply (drule_tac f = "%u. (1/2) *u" in arg_cong) |
|
950 |
apply (simp) |
|
951 |
done |
|
952 |
||
953 |
lemma Bolzano_bisect_diff: |
|
954 |
"a \<le> b ==> |
|
955 |
snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) = |
|
956 |
(b-a) / (2 ^ n)" |
|
957 |
apply (induct "n") |
|
958 |
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def) |
|
959 |
done |
|
960 |
||
961 |
lemmas Bolzano_nest_unique = |
|
962 |
lemma_nest_unique |
|
963 |
[OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le] |
|
964 |
||
965 |
||
966 |
lemma not_P_Bolzano_bisect: |
|
967 |
assumes P: "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)" |
|
968 |
and notP: "~ P(a,b)" |
|
969 |
and le: "a \<le> b" |
|
970 |
shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))" |
|
971 |
proof (induct n) |
|
972 |
case 0 thus ?case by simp |
|
973 |
next |
|
974 |
case (Suc n) |
|
975 |
thus ?case |
|
976 |
by (auto simp del: surjective_pairing [symmetric] |
|
977 |
simp add: Let_def split_def Bolzano_bisect_le [OF le] |
|
978 |
P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"]) |
|
979 |
qed |
|
980 |
||
981 |
(*Now we re-package P_prem as a formula*) |
|
982 |
lemma not_P_Bolzano_bisect': |
|
983 |
"[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c); |
|
984 |
~ P(a,b); a \<le> b |] ==> |
|
985 |
\<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))" |
|
986 |
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE]) |
|
987 |
||
988 |
||
989 |
||
990 |
lemma lemma_BOLZANO: |
|
991 |
"[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c); |
|
992 |
\<forall>x. \<exists>d::real. 0 < d & |
|
993 |
(\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b)); |
|
994 |
a \<le> b |] |
|
995 |
==> P(a,b)" |
|
996 |
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+) |
|
997 |
apply (rule LIMSEQ_minus_cancel) |
|
998 |
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero) |
|
999 |
apply (rule ccontr) |
|
1000 |
apply (drule not_P_Bolzano_bisect', assumption+) |
|
1001 |
apply (rename_tac "l") |
|
1002 |
apply (drule_tac x = l in spec, clarify) |
|
1003 |
apply (simp add: LIMSEQ_def) |
|
1004 |
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec) |
|
1005 |
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec) |
|
1006 |
apply (drule real_less_half_sum, auto) |
|
1007 |
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec) |
|
1008 |
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec) |
|
1009 |
apply safe |
|
1010 |
apply (simp_all (no_asm_simp)) |
|
1011 |
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) |
|
1012 |
apply (simp (no_asm_simp) add: abs_if) |
|
1013 |
apply (rule real_sum_of_halves [THEN subst]) |
|
1014 |
apply (rule add_strict_mono) |
|
1015 |
apply (simp_all add: diff_minus [symmetric]) |
|
1016 |
done |
|
1017 |
||
1018 |
||
1019 |
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) & |
|
1020 |
(\<forall>x. \<exists>d::real. 0 < d & |
|
1021 |
(\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b)))) |
|
1022 |
--> (\<forall>a b. a \<le> b --> P(a,b))" |
|
1023 |
apply clarify |
|
1024 |
apply (blast intro: lemma_BOLZANO) |
|
1025 |
done |
|
1026 |
||
1027 |
||
1028 |
subsection {* Intermediate Value Theorem *} |
|
1029 |
||
1030 |
text {*Prove Contrapositive by Bisection*} |
|
1031 |
||
1032 |
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b); |
|
1033 |
a \<le> b; |
|
1034 |
(\<forall>x. a \<le> x & x \<le> b --> isCont f x) |] |
|
1035 |
==> \<exists>x. a \<le> x & x \<le> b & f(x) = y" |
|
1036 |
apply (rule contrapos_pp, assumption) |
|
1037 |
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) |
|
1038 |
apply safe |
|
1039 |
apply simp_all |
|
1040 |
apply (simp add: isCont_iff LIM_def) |
|
1041 |
apply (rule ccontr) |
|
1042 |
apply (subgoal_tac "a \<le> x & x \<le> b") |
|
1043 |
prefer 2 |
|
1044 |
apply simp |
|
1045 |
apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith) |
|
1046 |
apply (drule_tac x = x in spec)+ |
|
1047 |
apply simp |
|
1048 |
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec) |
|
1049 |
apply safe |
|
1050 |
apply simp |
|
1051 |
apply (drule_tac x = s in spec, clarify) |
|
1052 |
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe) |
|
1053 |
apply (drule_tac x = "ba-x" in spec) |
|
1054 |
apply (simp_all add: abs_if) |
|
1055 |
apply (drule_tac x = "aa-x" in spec) |
|
1056 |
apply (case_tac "x \<le> aa", simp_all) |
|
1057 |
done |
|
1058 |
||
1059 |
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a); |
|
1060 |
a \<le> b; |
|
1061 |
(\<forall>x. a \<le> x & x \<le> b --> isCont f x) |
|
1062 |
|] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y" |
|
1063 |
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify) |
|
1064 |
apply (drule IVT [where f = "%x. - f x"], assumption) |
|
1065 |
apply (auto intro: isCont_minus) |
|
1066 |
done |
|
1067 |
||
1068 |
(*HOL style here: object-level formulations*) |
|
1069 |
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b & |
|
1070 |
(\<forall>x. a \<le> x & x \<le> b --> isCont f x)) |
|
1071 |
--> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)" |
|
1072 |
apply (blast intro: IVT) |
|
1073 |
done |
|
1074 |
||
1075 |
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b & |
|
1076 |
(\<forall>x. a \<le> x & x \<le> b --> isCont f x)) |
|
1077 |
--> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)" |
|
1078 |
apply (blast intro: IVT2) |
|
1079 |
done |
|
1080 |
||
1081 |
text{*By bisection, function continuous on closed interval is bounded above*} |
|
1082 |
||
1083 |
lemma isCont_bounded: |
|
1084 |
"[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |] |
|
1085 |
==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M" |
|
1086 |
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) |
|
1087 |
apply safe |
|
1088 |
apply simp_all |
|
1089 |
apply (rename_tac x xa ya M Ma) |
|
1090 |
apply (cut_tac x = M and y = Ma in linorder_linear, safe) |
|
1091 |
apply (rule_tac x = Ma in exI, clarify) |
|
1092 |
apply (cut_tac x = xb and y = xa in linorder_linear, force) |
|
1093 |
apply (rule_tac x = M in exI, clarify) |
|
1094 |
apply (cut_tac x = xb and y = xa in linorder_linear, force) |
|
1095 |
apply (case_tac "a \<le> x & x \<le> b") |
|
1096 |
apply (rule_tac [2] x = 1 in exI) |
|
1097 |
prefer 2 apply force |
|
1098 |
apply (simp add: LIM_def isCont_iff) |
|
1099 |
apply (drule_tac x = x in spec, auto) |
|
1100 |
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl) |
|
1101 |
apply (drule_tac x = 1 in spec, auto) |
|
1102 |
apply (rule_tac x = s in exI, clarify) |
|
1103 |
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify) |
|
1104 |
apply (drule_tac x = "xa-x" in spec) |
|
1105 |
apply (auto simp add: abs_ge_self) |
|
1106 |
done |
|
1107 |
||
1108 |
text{*Refine the above to existence of least upper bound*} |
|
1109 |
||
1110 |
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) --> |
|
1111 |
(\<exists>t. isLub UNIV S t)" |
|
1112 |
by (blast intro: reals_complete) |
|
1113 |
||
1114 |
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |] |
|
1115 |
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) & |
|
1116 |
(\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))" |
|
1117 |
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)" |
|
1118 |
in lemma_reals_complete) |
|
1119 |
apply auto |
|
1120 |
apply (drule isCont_bounded, assumption) |
|
1121 |
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def) |
|
1122 |
apply (rule exI, auto) |
|
1123 |
apply (auto dest!: spec simp add: linorder_not_less) |
|
1124 |
done |
|
1125 |
||
1126 |
text{*Now show that it attains its upper bound*} |
|
1127 |
||
1128 |
lemma isCont_eq_Ub: |
|
1129 |
assumes le: "a \<le> b" |
|
1130 |
and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x" |
|
1131 |
shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) & |
|
1132 |
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)" |
|
1133 |
proof - |
|
1134 |
from isCont_has_Ub [OF le con] |
|
1135 |
obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" |
|
1136 |
and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x" by blast |
|
1137 |
show ?thesis |
|
1138 |
proof (intro exI, intro conjI) |
|
1139 |
show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1) |
|
1140 |
show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M" |
|
1141 |
proof (rule ccontr) |
|
1142 |
assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)" |
|
1143 |
with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M" |
|
1144 |
by (fastsimp simp add: linorder_not_le [symmetric]) |
|
1145 |
hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x" |
|
1146 |
by (auto simp add: isCont_inverse isCont_diff con) |
|
1147 |
from isCont_bounded [OF le this] |
|
1148 |
obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto |
|
1149 |
have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))" |
|
1150 |
by (simp add: M3 compare_rls) |
|
1151 |
have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k |
|
1152 |
by (auto intro: order_le_less_trans [of _ k]) |
|
1153 |
with Minv |
|
1154 |
have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))" |
|
1155 |
by (intro strip less_imp_inverse_less, simp_all) |
|
1156 |
hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x" |
|
1157 |
by simp |
|
1158 |
have "M - inverse (k+1) < M" using k [of a] Minv [of a] le |
|
1159 |
by (simp, arith) |
|
1160 |
from M2 [OF this] |
|
1161 |
obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" .. |
|
1162 |
thus False using invlt [of x] by force |
|
1163 |
qed |
|
1164 |
qed |
|
1165 |
qed |
|
1166 |
||
1167 |
||
1168 |
text{*Same theorem for lower bound*} |
|
1169 |
||
1170 |
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |] |
|
1171 |
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) & |
|
1172 |
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)" |
|
1173 |
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x") |
|
1174 |
prefer 2 apply (blast intro: isCont_minus) |
|
1175 |
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub) |
|
1176 |
apply safe |
|
1177 |
apply auto |
|
1178 |
done |
|
1179 |
||
1180 |
||
1181 |
text{*Another version.*} |
|
1182 |
||
1183 |
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |] |
|
1184 |
==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) & |
|
1185 |
(\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))" |
|
1186 |
apply (frule isCont_eq_Lb) |
|
1187 |
apply (frule_tac [2] isCont_eq_Ub) |
|
1188 |
apply (assumption+, safe) |
|
1189 |
apply (rule_tac x = "f x" in exI) |
|
1190 |
apply (rule_tac x = "f xa" in exI, simp, safe) |
|
1191 |
apply (cut_tac x = x and y = xa in linorder_linear, safe) |
|
1192 |
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl) |
|
1193 |
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe) |
|
1194 |
apply (rule_tac [2] x = xb in exI) |
|
1195 |
apply (rule_tac [4] x = xb in exI, simp_all) |
|
1196 |
done |
|
1197 |
||
1198 |
||
1199 |
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*} |
|
1200 |
||
1201 |
lemma DERIV_left_inc: |
|
1202 |
fixes f :: "real => real" |
|
1203 |
assumes der: "DERIV f x :> l" |
|
1204 |
and l: "0 < l" |
|
1205 |
shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)" |
|
1206 |
proof - |
|
1207 |
from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]] |
|
1208 |
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)" |
|
1209 |
by (simp add: diff_minus) |
|
1210 |
then obtain s |
|
1211 |
where s: "0 < s" |
|
1212 |
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l" |
|
1213 |
by auto |
|
1214 |
thus ?thesis |
|
1215 |
proof (intro exI conjI strip) |
|
1216 |
show "0<s" . |
|
1217 |
fix h::real |
|
1218 |
assume "0 < h" "h < s" |
|
1219 |
with all [of h] show "f x < f (x+h)" |
|
1220 |
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric] |
|
1221 |
split add: split_if_asm) |
|
1222 |
assume "~ (f (x+h) - f x) / h < l" and h: "0 < h" |
|
1223 |
with l |
|
1224 |
have "0 < (f (x+h) - f x) / h" by arith |
|
1225 |
thus "f x < f (x+h)" |
|
1226 |
by (simp add: pos_less_divide_eq h) |
|
1227 |
qed |
|
1228 |
qed |
|
1229 |
qed |
|
1230 |
||
1231 |
lemma DERIV_left_dec: |
|
1232 |
fixes f :: "real => real" |
|
1233 |
assumes der: "DERIV f x :> l" |
|
1234 |
and l: "l < 0" |
|
1235 |
shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)" |
|
1236 |
proof - |
|
1237 |
from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]] |
|
1238 |
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)" |
|
1239 |
by (simp add: diff_minus) |
|
1240 |
then obtain s |
|
1241 |
where s: "0 < s" |
|
1242 |
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l" |
|
1243 |
by auto |
|
1244 |
thus ?thesis |
|
1245 |
proof (intro exI conjI strip) |
|
1246 |
show "0<s" . |
|
1247 |
fix h::real |
|
1248 |
assume "0 < h" "h < s" |
|
1249 |
with all [of "-h"] show "f x < f (x-h)" |
|
1250 |
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric] |
|
1251 |
split add: split_if_asm) |
|
1252 |
assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h" |
|
1253 |
with l |
|
1254 |
have "0 < (f (x-h) - f x) / h" by arith |
|
1255 |
thus "f x < f (x-h)" |
|
1256 |
by (simp add: pos_less_divide_eq h) |
|
1257 |
qed |
|
1258 |
qed |
|
1259 |
qed |
|
1260 |
||
1261 |
lemma DERIV_local_max: |
|
1262 |
fixes f :: "real => real" |
|
1263 |
assumes der: "DERIV f x :> l" |
|
1264 |
and d: "0 < d" |
|
1265 |
and le: "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)" |
|
1266 |
shows "l = 0" |
|
1267 |
proof (cases rule: linorder_cases [of l 0]) |
|
1268 |
case equal show ?thesis . |
|
1269 |
next |
|
1270 |
case less |
|
1271 |
from DERIV_left_dec [OF der less] |
|
1272 |
obtain d' where d': "0 < d'" |
|
1273 |
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast |
|
1274 |
from real_lbound_gt_zero [OF d d'] |
|
1275 |
obtain e where "0 < e \<and> e < d \<and> e < d'" .. |
|
1276 |
with lt le [THEN spec [where x="x-e"]] |
|
1277 |
show ?thesis by (auto simp add: abs_if) |
|
1278 |
next |
|
1279 |
case greater |
|
1280 |
from DERIV_left_inc [OF der greater] |
|
1281 |
obtain d' where d': "0 < d'" |
|
1282 |
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast |
|
1283 |
from real_lbound_gt_zero [OF d d'] |
|
1284 |
obtain e where "0 < e \<and> e < d \<and> e < d'" .. |
|
1285 |
with lt le [THEN spec [where x="x+e"]] |
|
1286 |
show ?thesis by (auto simp add: abs_if) |
|
1287 |
qed |
|
1288 |
||
1289 |
||
1290 |
text{*Similar theorem for a local minimum*} |
|
1291 |
lemma DERIV_local_min: |
|
1292 |
fixes f :: "real => real" |
|
1293 |
shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0" |
|
1294 |
by (drule DERIV_minus [THEN DERIV_local_max], auto) |
|
1295 |
||
1296 |
||
1297 |
text{*In particular, if a function is locally flat*} |
|
1298 |
lemma DERIV_local_const: |
|
1299 |
fixes f :: "real => real" |
|
1300 |
shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0" |
|
1301 |
by (auto dest!: DERIV_local_max) |
|
1302 |
||
1303 |
text{*Lemma about introducing open ball in open interval*} |
|
1304 |
lemma lemma_interval_lt: |
|
1305 |
"[| a < x; x < b |] |
|
1306 |
==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)" |
|
1307 |
apply (simp add: abs_interval_iff) |
|
1308 |
apply (insert linorder_linear [of "x-a" "b-x"], safe) |
|
1309 |
apply (rule_tac x = "x-a" in exI) |
|
1310 |
apply (rule_tac [2] x = "b-x" in exI, auto) |
|
1311 |
done |
|
1312 |
||
1313 |
lemma lemma_interval: "[| a < x; x < b |] ==> |
|
1314 |
\<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)" |
|
1315 |
apply (drule lemma_interval_lt, auto) |
|
1316 |
apply (auto intro!: exI) |
|
1317 |
done |
|
1318 |
||
1319 |
text{*Rolle's Theorem. |
|
1320 |
If @{term f} is defined and continuous on the closed interval |
|
1321 |
@{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"}, |
|
1322 |
and @{term "f(a) = f(b)"}, |
|
1323 |
then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*} |
|
1324 |
theorem Rolle: |
|
1325 |
assumes lt: "a < b" |
|
1326 |
and eq: "f(a) = f(b)" |
|
1327 |
and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x" |
|
1328 |
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
|
1329 |
shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0" |
21164 | 1330 |
proof - |
1331 |
have le: "a \<le> b" using lt by simp |
|
1332 |
from isCont_eq_Ub [OF le con] |
|
1333 |
obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x" |
|
1334 |
and alex: "a \<le> x" and xleb: "x \<le> b" |
|
1335 |
by blast |
|
1336 |
from isCont_eq_Lb [OF le con] |
|
1337 |
obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z" |
|
1338 |
and alex': "a \<le> x'" and x'leb: "x' \<le> b" |
|
1339 |
by blast |
|
1340 |
show ?thesis |
|
1341 |
proof cases |
|
1342 |
assume axb: "a < x & x < b" |
|
1343 |
--{*@{term f} attains its maximum within the interval*} |
|
1344 |
hence ax: "a<x" and xb: "x<b" by auto |
|
1345 |
from lemma_interval [OF ax xb] |
|
1346 |
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" |
|
1347 |
by blast |
|
1348 |
hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max |
|
1349 |
by blast |
|
1350 |
from differentiableD [OF dif [OF axb]] |
|
1351 |
obtain l where der: "DERIV f x :> l" .. |
|
1352 |
have "l=0" by (rule DERIV_local_max [OF der d bound']) |
|
1353 |
--{*the derivative at a local maximum is zero*} |
|
1354 |
thus ?thesis using ax xb der by auto |
|
1355 |
next |
|
1356 |
assume notaxb: "~ (a < x & x < b)" |
|
1357 |
hence xeqab: "x=a | x=b" using alex xleb by arith |
|
1358 |
hence fb_eq_fx: "f b = f x" by (auto simp add: eq) |
|
1359 |
show ?thesis |
|
1360 |
proof cases |
|
1361 |
assume ax'b: "a < x' & x' < b" |
|
1362 |
--{*@{term f} attains its minimum within the interval*} |
|
1363 |
hence ax': "a<x'" and x'b: "x'<b" by auto |
|
1364 |
from lemma_interval [OF ax' x'b] |
|
1365 |
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" |
|
1366 |
by blast |
|
1367 |
hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min |
|
1368 |
by blast |
|
1369 |
from differentiableD [OF dif [OF ax'b]] |
|
1370 |
obtain l where der: "DERIV f x' :> l" .. |
|
1371 |
have "l=0" by (rule DERIV_local_min [OF der d bound']) |
|
1372 |
--{*the derivative at a local minimum is zero*} |
|
1373 |
thus ?thesis using ax' x'b der by auto |
|
1374 |
next |
|
1375 |
assume notax'b: "~ (a < x' & x' < b)" |
|
1376 |
--{*@{term f} is constant througout the interval*} |
|
1377 |
hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith |
|
1378 |
hence fb_eq_fx': "f b = f x'" by (auto simp add: eq) |
|
1379 |
from dense [OF lt] |
|
1380 |
obtain r where ar: "a < r" and rb: "r < b" by blast |
|
1381 |
from lemma_interval [OF ar rb] |
|
1382 |
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b" |
|
1383 |
by blast |
|
1384 |
have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b" |
|
1385 |
proof (clarify) |
|
1386 |
fix z::real |
|
1387 |
assume az: "a \<le> z" and zb: "z \<le> b" |
|
1388 |
show "f z = f b" |
|
1389 |
proof (rule order_antisym) |
|
1390 |
show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb) |
|
1391 |
show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb) |
|
1392 |
qed |
|
1393 |
qed |
|
1394 |
have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y" |
|
1395 |
proof (intro strip) |
|
1396 |
fix y::real |
|
1397 |
assume lt: "\<bar>r-y\<bar> < d" |
|
1398 |
hence "f y = f b" by (simp add: eq_fb bound) |
|
1399 |
thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le) |
|
1400 |
qed |
|
1401 |
from differentiableD [OF dif [OF conjI [OF ar rb]]] |
|
1402 |
obtain l where der: "DERIV f r :> l" .. |
|
1403 |
have "l=0" by (rule DERIV_local_const [OF der d bound']) |
|
1404 |
--{*the derivative of a constant function is zero*} |
|
1405 |
thus ?thesis using ar rb der by auto |
|
1406 |
qed |
|
1407 |
qed |
|
1408 |
qed |
|
1409 |
||
1410 |
||
1411 |
subsection{*Mean Value Theorem*} |
|
1412 |
||
1413 |
lemma lemma_MVT: |
|
1414 |
"f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)" |
|
1415 |
proof cases |
|
1416 |
assume "a=b" thus ?thesis by simp |
|
1417 |
next |
|
1418 |
assume "a\<noteq>b" |
|
1419 |
hence ba: "b-a \<noteq> 0" by arith |
|
1420 |
show ?thesis |
|
1421 |
by (rule real_mult_left_cancel [OF ba, THEN iffD1], |
|
1422 |
simp add: right_diff_distrib, |
|
1423 |
simp add: left_diff_distrib) |
|
1424 |
qed |
|
1425 |
||
1426 |
theorem MVT: |
|
1427 |
assumes lt: "a < b" |
|
1428 |
and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x" |
|
1429 |
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
|
1430 |
shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l & |
21164 | 1431 |
(f(b) - f(a) = (b-a) * l)" |
1432 |
proof - |
|
1433 |
let ?F = "%x. f x - ((f b - f a) / (b-a)) * x" |
|
1434 |
have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con |
|
1435 |
by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id) |
|
1436 |
have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x" |
|
1437 |
proof (clarify) |
|
1438 |
fix x::real |
|
1439 |
assume ax: "a < x" and xb: "x < b" |
|
1440 |
from differentiableD [OF dif [OF conjI [OF ax xb]]] |
|
1441 |
obtain l where der: "DERIV f x :> l" .. |
|
1442 |
show "?F differentiable x" |
|
1443 |
by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"], |
|
1444 |
blast intro: DERIV_diff DERIV_cmult_Id der) |
|
1445 |
qed |
|
1446 |
from Rolle [where f = ?F, OF lt lemma_MVT contF difF] |
|
1447 |
obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0" |
|
1448 |
by blast |
|
1449 |
have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)" |
|
1450 |
by (rule DERIV_cmult_Id) |
|
1451 |
hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z |
|
1452 |
:> 0 + (f b - f a) / (b - a)" |
|
1453 |
by (rule DERIV_add [OF der]) |
|
1454 |
show ?thesis |
|
1455 |
proof (intro exI conjI) |
|
1456 |
show "a < z" . |
|
1457 |
show "z < b" . |
|
1458 |
show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp) |
|
1459 |
show "DERIV f z :> ((f b - f a)/(b-a))" using derF by simp |
|
1460 |
qed |
|
1461 |
qed |
|
1462 |
||
1463 |
||
1464 |
text{*A function is constant if its derivative is 0 over an interval.*} |
|
1465 |
||
1466 |
lemma DERIV_isconst_end: |
|
1467 |
fixes f :: "real => real" |
|
1468 |
shows "[| a < b; |
|
1469 |
\<forall>x. a \<le> x & x \<le> b --> isCont f x; |
|
1470 |
\<forall>x. a < x & x < b --> DERIV f x :> 0 |] |
|
1471 |
==> f b = f a" |
|
1472 |
apply (drule MVT, assumption) |
|
1473 |
apply (blast intro: differentiableI) |
|
1474 |
apply (auto dest!: DERIV_unique simp add: diff_eq_eq) |
|
1475 |
done |
|
1476 |
||
1477 |
lemma DERIV_isconst1: |
|
1478 |
fixes f :: "real => real" |
|
1479 |
shows "[| a < b; |
|
1480 |
\<forall>x. a \<le> x & x \<le> b --> isCont f x; |
|
1481 |
\<forall>x. a < x & x < b --> DERIV f x :> 0 |] |
|
1482 |
==> \<forall>x. a \<le> x & x \<le> b --> f x = f a" |
|
1483 |
apply safe |
|
1484 |
apply (drule_tac x = a in order_le_imp_less_or_eq, safe) |
|
1485 |
apply (drule_tac b = x in DERIV_isconst_end, auto) |
|
1486 |
done |
|
1487 |
||
1488 |
lemma DERIV_isconst2: |
|
1489 |
fixes f :: "real => real" |
|
1490 |
shows "[| a < b; |
|
1491 |
\<forall>x. a \<le> x & x \<le> b --> isCont f x; |
|
1492 |
\<forall>x. a < x & x < b --> DERIV f x :> 0; |
|
1493 |
a \<le> x; x \<le> b |] |
|
1494 |
==> f x = f a" |
|
1495 |
apply (blast dest: DERIV_isconst1) |
|
1496 |
done |
|
1497 |
||
1498 |
lemma DERIV_isconst_all: |
|
1499 |
fixes f :: "real => real" |
|
1500 |
shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)" |
|
1501 |
apply (rule linorder_cases [of x y]) |
|
1502 |
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+ |
|
1503 |
done |
|
1504 |
||
1505 |
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
|
1506 |
fixes f :: "real => real" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
1507 |
shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k" |
21164 | 1508 |
apply (rule linorder_cases [of a b], auto) |
1509 |
apply (drule_tac [!] f = f in MVT) |
|
1510 |
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def) |
|
1511 |
apply (auto dest: DERIV_unique simp add: left_distrib diff_minus) |
|
1512 |
done |
|
1513 |
||
1514 |
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
|
1515 |
fixes f :: "real => real" |
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
1516 |
shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k" |
21164 | 1517 |
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1]) |
1518 |
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc) |
|
1519 |
done |
|
1520 |
||
1521 |
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)" |
|
1522 |
by (simp) |
|
1523 |
||
1524 |
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)" |
|
1525 |
by (simp) |
|
1526 |
||
1527 |
text{*Gallileo's "trick": average velocity = av. of end velocities*} |
|
1528 |
||
1529 |
lemma DERIV_const_average: |
|
1530 |
fixes v :: "real => real" |
|
1531 |
assumes neq: "a \<noteq> (b::real)" |
|
1532 |
and der: "\<forall>x. DERIV v x :> k" |
|
1533 |
shows "v ((a + b)/2) = (v a + v b)/2" |
|
1534 |
proof (cases rule: linorder_cases [of a b]) |
|
1535 |
case equal with neq show ?thesis by simp |
|
1536 |
next |
|
1537 |
case less |
|
1538 |
have "(v b - v a) / (b - a) = k" |
|
1539 |
by (rule DERIV_const_ratio_const2 [OF neq der]) |
|
1540 |
hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp |
|
1541 |
moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k" |
|
1542 |
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq) |
|
1543 |
ultimately show ?thesis using neq by force |
|
1544 |
next |
|
1545 |
case greater |
|
1546 |
have "(v b - v a) / (b - a) = k" |
|
1547 |
by (rule DERIV_const_ratio_const2 [OF neq der]) |
|
1548 |
hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp |
|
1549 |
moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k" |
|
1550 |
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq) |
|
1551 |
ultimately show ?thesis using neq by (force simp add: add_commute) |
|
1552 |
qed |
|
1553 |
||
1554 |
||
1555 |
text{*Dull lemma: an continuous injection on an interval must have a |
|
1556 |
strict maximum at an end point, not in the middle.*} |
|
1557 |
||
1558 |
lemma lemma_isCont_inj: |
|
1559 |
fixes f :: "real \<Rightarrow> real" |
|
1560 |
assumes d: "0 < d" |
|
1561 |
and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z" |
|
1562 |
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z" |
|
1563 |
shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z" |
|
1564 |
proof (rule ccontr) |
|
1565 |
assume "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)" |
|
1566 |
hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto |
|
1567 |
show False |
|
1568 |
proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"]) |
|
1569 |
case le |
|
1570 |
from d cont all [of "x+d"] |
|
1571 |
have flef: "f(x+d) \<le> f x" |
|
1572 |
and xlex: "x - d \<le> x" |
|
1573 |
and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z" |
|
1574 |
by (auto simp add: abs_if) |
|
1575 |
from IVT [OF le flef xlex cont'] |
|
1576 |
obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast |
|
1577 |
moreover |
|
1578 |
hence "g(f x') = g (f(x+d))" by simp |
|
1579 |
ultimately show False using d inj [of x'] inj [of "x+d"] |
|
1580 |
by (simp add: abs_le_interval_iff) |
|
1581 |
next |
|
1582 |
case ge |
|
1583 |
from d cont all [of "x-d"] |
|
1584 |
have flef: "f(x-d) \<le> f x" |
|
1585 |
and xlex: "x \<le> x+d" |
|
1586 |
and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" |
|
1587 |
by (auto simp add: abs_if) |
|
1588 |
from IVT2 [OF ge flef xlex cont'] |
|
1589 |
obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast |
|
1590 |
moreover |
|
1591 |
hence "g(f x') = g (f(x-d))" by simp |
|
1592 |
ultimately show False using d inj [of x'] inj [of "x-d"] |
|
1593 |
by (simp add: abs_le_interval_iff) |
|
1594 |
qed |
|
1595 |
qed |
|
1596 |
||
1597 |
||
1598 |
text{*Similar version for lower bound.*} |
|
1599 |
||
1600 |
lemma lemma_isCont_inj2: |
|
1601 |
fixes f g :: "real \<Rightarrow> real" |
|
1602 |
shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z; |
|
1603 |
\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |] |
|
1604 |
==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x" |
|
1605 |
apply (insert lemma_isCont_inj |
|
1606 |
[where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d]) |
|
1607 |
apply (simp add: isCont_minus linorder_not_le) |
|
1608 |
done |
|
1609 |
||
1610 |
text{*Show there's an interval surrounding @{term "f(x)"} in |
|
1611 |
@{text "f[[x - d, x + d]]"} .*} |
|
1612 |
||
1613 |
lemma isCont_inj_range: |
|
1614 |
fixes f :: "real \<Rightarrow> real" |
|
1615 |
assumes d: "0 < d" |
|
1616 |
and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z" |
|
1617 |
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z" |
|
1618 |
shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)" |
|
1619 |
proof - |
|
1620 |
have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d |
|
1621 |
by (auto simp add: abs_le_interval_iff) |
|
1622 |
from isCont_Lb_Ub [OF this] |
|
1623 |
obtain L M |
|
1624 |
where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M" |
|
1625 |
and all2 [rule_format]: |
|
1626 |
"\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)" |
|
1627 |
by auto |
|
1628 |
with d have "L \<le> f x & f x \<le> M" by simp |
|
1629 |
moreover have "L \<noteq> f x" |
|
1630 |
proof - |
|
1631 |
from lemma_isCont_inj2 [OF d inj cont] |
|
1632 |
obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x" by auto |
|
1633 |
thus ?thesis using all1 [of u] by arith |
|
1634 |
qed |
|
1635 |
moreover have "f x \<noteq> M" |
|
1636 |
proof - |
|
1637 |
from lemma_isCont_inj [OF d inj cont] |
|
1638 |
obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u" by auto |
|
1639 |
thus ?thesis using all1 [of u] by arith |
|
1640 |
qed |
|
1641 |
ultimately have "L < f x & f x < M" by arith |
|
1642 |
hence "0 < f x - L" "0 < M - f x" by arith+ |
|
1643 |
from real_lbound_gt_zero [OF this] |
|
1644 |
obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto |
|
1645 |
thus ?thesis |
|
1646 |
proof (intro exI conjI) |
|
1647 |
show "0<e" . |
|
1648 |
show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)" |
|
1649 |
proof (intro strip) |
|
1650 |
fix y::real |
|
1651 |
assume "\<bar>y - f x\<bar> \<le> e" |
|
1652 |
with e have "L \<le> y \<and> y \<le> M" by arith |
|
1653 |
from all2 [OF this] |
|
1654 |
obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast |
|
1655 |
thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y" |
|
1656 |
by (force simp add: abs_le_interval_iff) |
|
1657 |
qed |
|
1658 |
qed |
|
1659 |
qed |
|
1660 |
||
1661 |
||
1662 |
text{*Continuity of inverse function*} |
|
1663 |
||
1664 |
lemma isCont_inverse_function: |
|
1665 |
fixes f g :: "real \<Rightarrow> real" |
|
1666 |
assumes d: "0 < d" |
|
1667 |
and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z" |
|
1668 |
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z" |
|
1669 |
shows "isCont g (f x)" |
|
1670 |
proof (simp add: isCont_iff LIM_eq) |
|
1671 |
show "\<forall>r. 0 < r \<longrightarrow> |
|
1672 |
(\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)" |
|
1673 |
proof (intro strip) |
|
1674 |
fix r::real |
|
1675 |
assume r: "0<r" |
|
1676 |
from real_lbound_gt_zero [OF r d] |
|
1677 |
obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast |
|
1678 |
with inj cont |
|
1679 |
have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z" |
|
1680 |
"\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z" by auto |
|
1681 |
from isCont_inj_range [OF e this] |
|
1682 |
obtain e' where e': "0 < e'" |
|
1683 |
and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)" |
|
1684 |
by blast |
|
1685 |
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" |
|
1686 |
proof (intro exI conjI) |
|
1687 |
show "0<e'" . |
|
1688 |
show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r" |
|
1689 |
proof (intro strip) |
|
1690 |
fix z::real |
|
1691 |
assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'" |
|
1692 |
with e e_lt e_simps all [rule_format, of "f x + z"] |
|
1693 |
show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force |
|
1694 |
qed |
|
1695 |
qed |
|
1696 |
qed |
|
1697 |
qed |
|
1698 |
||
1699 |
theorem GMVT: |
|
21784
e76faa6e65fd
changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents:
21404
diff
changeset
|
1700 |
fixes a b :: real |
21164 | 1701 |
assumes alb: "a < b" |
1702 |
and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x" |
|
1703 |
and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x" |
|
1704 |
and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x" |
|
1705 |
and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x" |
|
1706 |
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)" |
|
1707 |
proof - |
|
1708 |
let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)" |
|
1709 |
from prems have "a < b" by simp |
|
1710 |
moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x" |
|
1711 |
proof - |
|
1712 |
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp |
|
1713 |
with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x" |
|
1714 |
by (auto intro: isCont_mult) |
|
1715 |
moreover |
|
1716 |
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp |
|
1717 |
with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x" |
|
1718 |
by (auto intro: isCont_mult) |
|
1719 |
ultimately show ?thesis |
|
1720 |
by (fastsimp intro: isCont_diff) |
|
1721 |
qed |
|
1722 |
moreover |
|
1723 |
have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x" |
|
1724 |
proof - |
|
1725 |
have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const) |
|
1726 |
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) |
|
1727 |
moreover |
|
1728 |
have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const) |
|
1729 |
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) |
|
1730 |
ultimately show ?thesis by (simp add: differentiable_diff) |
|
1731 |
qed |
|
1732 |
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) |
|
1733 |
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" .. |
|
1734 |
then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" .. |
|
1735 |
||
1736 |
from cdef have cint: "a < c \<and> c < b" by auto |
|
1737 |
with gd have "g differentiable c" by simp |
|
1738 |
hence "\<exists>D. DERIV g c :> D" by (rule differentiableD) |
|
1739 |
then obtain g'c where g'cdef: "DERIV g c :> g'c" .. |
|
1740 |
||
1741 |
from cdef have "a < c \<and> c < b" by auto |
|
1742 |
with fd have "f differentiable c" by simp |
|
1743 |
hence "\<exists>D. DERIV f c :> D" by (rule differentiableD) |
|
1744 |
then obtain f'c where f'cdef: "DERIV f c :> f'c" .. |
|
1745 |
||
1746 |
from cdef have "DERIV ?h c :> l" by auto |
|
1747 |
moreover |
|
1748 |
{ |
|
1749 |
from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)" |
|
1750 |
apply (insert DERIV_const [where k="f b - f a"]) |
|
1751 |
apply (drule meta_spec [of _ c]) |
|
1752 |
apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g]) |
|
1753 |
by simp_all |
|
1754 |
moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)" |
|
1755 |
apply (insert DERIV_const [where k="g b - g a"]) |
|
1756 |
apply (drule meta_spec [of _ c]) |
|
1757 |
apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f]) |
|
1758 |
by simp_all |
|
1759 |
ultimately have "DERIV ?h c :> g'c * (f b - f a) - f'c * (g b - g a)" |
|
1760 |
by (simp add: DERIV_diff) |
|
1761 |
} |
|
1762 |
ultimately have leq: "l = g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique) |
|
1763 |
||
1764 |
{ |
|
1765 |
from cdef have "?h b - ?h a = (b - a) * l" by auto |
|
1766 |
also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp |
|
1767 |
finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp |
|
1768 |
} |
|
1769 |
moreover |
|
1770 |
{ |
|
1771 |
have "?h b - ?h a = |
|
1772 |
((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) - |
|
1773 |
((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))" |
|
1774 |
by (simp add: mult_ac add_ac real_diff_mult_distrib) |
|
1775 |
hence "?h b - ?h a = 0" by auto |
|
1776 |
} |
|
1777 |
ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto |
|
1778 |
with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp |
|
1779 |
hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp |
|
1780 |
hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac) |
|
1781 |
||
1782 |
with g'cdef f'cdef cint show ?thesis by auto |
|
1783 |
qed |
|
1784 |
||
1785 |
end |