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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 :: "[real \<Rightarrow> 'a::real_normed_vector, real, '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)
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"DERIV f x :> D = ((%h. (f(x + h) - f x) /# h) -- 0 --> D)"
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nsderiv :: "[real=>real,real,real] => bool"
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("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
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"NSDERIV f x :> D = (\<forall>h \<in> Infinitesimal - {0}.
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(( *f* f)(hypreal_of_real x + h)
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- hypreal_of_real (f x))/h @= hypreal_of_real D)"
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differentiable :: "[real=>real,real] => bool" (infixl "differentiable" 60)
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"f differentiable x = (\<exists>D. DERIV f x :> D)"
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NSdifferentiable :: "[real=>real,real] => bool"
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(infixl "NSdifferentiable" 60)
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"f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)"
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increment :: "[real=>real,real,hypreal] => hypreal"
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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 real_scaleR_def 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 scaleR_right_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 scaleR_minus_right 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. h *# ((f(x+h) - f(x)) /# h)) -- 0 --> 0 *# D"
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by (intro LIM_scaleR LIM_self)
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hence "(\<lambda>h. (f(x+h) - f(x))) -- 0 --> 0"
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by (simp cong: LIM_cong)
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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_algebra"
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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 scaleR_right_distrib [symmetric] ring_distrib)
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lemma DERIV_mult':
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fixes f g :: "real \<Rightarrow> 'a::real_normed_algebra"
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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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fixes f g :: "real \<Rightarrow> 'a::{real_normed_algebra,comm_ring}" shows
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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::real. (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_LIM_iff':
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"((%h::real. (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_div_algebra)\<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 LIM_equal2:
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assumes 1: "0 < R"
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assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
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shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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apply (unfold LIM_def, safe)
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apply (drule_tac x="r" in spec, safe)
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apply (rule_tac x="min s R" in exI, safe)
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apply (simp add: 1)
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apply (simp add: 2)
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done
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lemma DERIV_inverse':
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fixes f :: "real \<Rightarrow> 'a::real_normed_div_algebra"
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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 :: real
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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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fixes D E :: "'a::{real_normed_div_algebra,field}"
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shows "\<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 :: "real \<Rightarrow> 'a::{real_normed_algebra,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
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show ?case by (simp add: power_Suc f)
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case (Suc k)
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from DERIV_mult' [OF f Suc] show ?case
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apply (simp only: of_nat_Suc scaleR_left_distrib scaleR_one)
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apply (simp only: power_Suc right_distrib mult_scaleR_right mult_ac)
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done
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qed
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lemma DERIV_power:
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fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower}"
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assumes f: "DERIV f x :> D"
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shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n *# (D * f x ^ (n - Suc 0))"
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by (cases "n", simp, simp add: DERIV_power_Suc f)
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(* ------------------------------------------------------------------------ *)
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(* Caratheodory formulation of derivative at a point: standard proof *)
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(* ------------------------------------------------------------------------ *)
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lemma CARAT_DERIV:
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"(DERIV f x :> l) =
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(\<exists>g. (\<forall>z. f z - f x = (z-x) *# g z) & isCont g x & g x = l)"
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(is "?lhs = ?rhs")
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proof
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assume der: "DERIV f x :> l"
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show "\<exists>g. (\<forall>z. f z - f x = (z-x) *# g z) \<and> isCont g x \<and> g x = l"
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proof (intro exI conjI)
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let ?g = "(%z. if z = x then l else (f z - f x) /# (z-x))"
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show "\<forall>z. f z - f x = (z-x) *# ?g z" by (simp)
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show "isCont ?g x" using der
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by (simp add: isCont_iff DERIV_iff diff_minus
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cong: LIM_equal [rule_format])
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show "?g x = l" by simp
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qed
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next
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assume "?rhs"
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then obtain g where
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"(\<forall>z. f z - f x = (z-x) *# g z)" and "isCont g x" and "g x = l" by blast
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thus "(DERIV f x :> l)"
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by (auto simp add: isCont_iff DERIV_iff diff_minus
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cong: LIM_equal [rule_format])
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qed
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lemma DERIV_chain':
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assumes f: "DERIV f x :> D"
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assumes g: "DERIV g (f x) :> E"
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shows "DERIV (\<lambda>x. g (f x)) x :> D *# E"
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proof (unfold DERIV_iff2)
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obtain d where d: "\<forall>y. g y - g (f x) = (y - f x) *# d y"
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and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
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using CARAT_DERIV [THEN iffD1, OF g] by fast
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from f have "f -- x --> f x"
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by (rule DERIV_isCont [unfolded isCont_def])
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with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
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by (rule LIM_compose)
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hence "(\<lambda>z. ((f z - f x) /# (z - x)) *# d (f z))
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-- x --> D *# d (f x)"
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by (rule LIM_scaleR [OF f [unfolded DERIV_iff2]])
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thus "(\<lambda>z. (g (f z) - g (f x)) /# (z - x)) -- x --> D *# E"
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by (simp add: d dfx real_scaleR_def)
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qed
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subsubsection {* Nonstandard proofs *}
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lemma DERIV_NS_iff:
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"(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/#h) -- 0 --NS> D)"
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by (simp add: deriv_def LIM_NSLIM_iff)
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lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --NS> D"
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by (simp add: deriv_def LIM_NSLIM_iff)
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|
321 |
lemma NSDeriv_unique:
|
|
322 |
"[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
|
|
323 |
apply (simp add: nsderiv_def)
|
|
324 |
apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
|
|
325 |
apply (auto dest!: bspec [where x=epsilon]
|
|
326 |
intro!: inj_hypreal_of_real [THEN injD]
|
|
327 |
dest: approx_trans3)
|
|
328 |
done
|
|
329 |
|
|
330 |
text {*First NSDERIV in terms of NSLIM*}
|
|
331 |
|
|
332 |
text{*first equivalence *}
|
|
333 |
lemma NSDERIV_NSLIM_iff:
|
|
334 |
"(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
|
|
335 |
apply (simp add: nsderiv_def NSLIM_def, auto)
|
|
336 |
apply (drule_tac x = xa in bspec)
|
|
337 |
apply (rule_tac [3] ccontr)
|
|
338 |
apply (drule_tac [3] x = h in spec)
|
|
339 |
apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
|
|
340 |
done
|
|
341 |
|
|
342 |
text{*second equivalence *}
|
|
343 |
lemma NSDERIV_NSLIM_iff2:
|
|
344 |
"(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
|
|
345 |
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff diff_minus [symmetric]
|
|
346 |
LIM_NSLIM_iff [symmetric])
|
|
347 |
|
|
348 |
(* while we're at it! *)
|
|
349 |
lemma NSDERIV_iff2:
|
|
350 |
"(NSDERIV f x :> D) =
|
|
351 |
(\<forall>w.
|
|
352 |
w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
|
|
353 |
( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
|
|
354 |
by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
|
|
355 |
|
|
356 |
(*FIXME DELETE*)
|
|
357 |
lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x - a \<noteq> (0::hypreal))"
|
|
358 |
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
|
|
359 |
|
|
360 |
lemma NSDERIVD5:
|
|
361 |
"(NSDERIV f x :> D) ==>
|
|
362 |
(\<forall>u. u \<approx> hypreal_of_real x -->
|
|
363 |
( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
|
|
364 |
apply (auto simp add: NSDERIV_iff2)
|
|
365 |
apply (case_tac "u = hypreal_of_real x", auto)
|
|
366 |
apply (drule_tac x = u in spec, auto)
|
|
367 |
apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
|
|
368 |
apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
|
|
369 |
apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
|
|
370 |
apply (auto simp add:
|
|
371 |
approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
|
|
372 |
Infinitesimal_subset_HFinite [THEN subsetD])
|
|
373 |
done
|
|
374 |
|
|
375 |
lemma NSDERIVD4:
|
|
376 |
"(NSDERIV f x :> D) ==>
|
|
377 |
(\<forall>h \<in> Infinitesimal.
|
|
378 |
(( *f* f)(hypreal_of_real x + h) -
|
|
379 |
hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
|
|
380 |
apply (auto simp add: nsderiv_def)
|
|
381 |
apply (case_tac "h = (0::hypreal) ")
|
|
382 |
apply (auto simp add: diff_minus)
|
|
383 |
apply (drule_tac x = h in bspec)
|
|
384 |
apply (drule_tac [2] c = h in approx_mult1)
|
|
385 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
|
|
386 |
simp add: diff_minus)
|
|
387 |
done
|
|
388 |
|
|
389 |
lemma NSDERIVD3:
|
|
390 |
"(NSDERIV f x :> D) ==>
|
|
391 |
(\<forall>h \<in> Infinitesimal - {0}.
|
|
392 |
(( *f* f)(hypreal_of_real x + h) -
|
|
393 |
hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
|
|
394 |
apply (auto simp add: nsderiv_def)
|
|
395 |
apply (rule ccontr, drule_tac x = h in bspec)
|
|
396 |
apply (drule_tac [2] c = h in approx_mult1)
|
|
397 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
|
|
398 |
simp add: mult_assoc diff_minus)
|
|
399 |
done
|
|
400 |
|
|
401 |
text{*Differentiability implies continuity
|
|
402 |
nice and simple "algebraic" proof*}
|
|
403 |
lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
|
|
404 |
apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
|
|
405 |
apply (drule approx_minus_iff [THEN iffD1])
|
|
406 |
apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
|
|
407 |
apply (drule_tac x = "xa - hypreal_of_real x" in bspec)
|
|
408 |
prefer 2 apply (simp add: add_assoc [symmetric])
|
|
409 |
apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
|
|
410 |
apply (drule_tac c = "xa - hypreal_of_real x" in approx_mult1)
|
|
411 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
|
|
412 |
simp add: mult_assoc)
|
|
413 |
apply (drule_tac x3=D in
|
|
414 |
HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
|
|
415 |
THEN mem_infmal_iff [THEN iffD1]])
|
|
416 |
apply (auto simp add: mult_commute
|
|
417 |
intro: approx_trans approx_minus_iff [THEN iffD2])
|
|
418 |
done
|
|
419 |
|
|
420 |
text{*Differentiation rules for combinations of functions
|
|
421 |
follow from clear, straightforard, algebraic
|
|
422 |
manipulations*}
|
|
423 |
text{*Constant function*}
|
|
424 |
|
|
425 |
(* use simple constant nslimit theorem *)
|
|
426 |
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
|
|
427 |
by (simp add: NSDERIV_NSLIM_iff)
|
|
428 |
|
|
429 |
text{*Sum of functions- proved easily*}
|
|
430 |
|
|
431 |
lemma NSDERIV_add: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
|
|
432 |
==> NSDERIV (%x. f x + g x) x :> Da + Db"
|
|
433 |
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
|
|
434 |
apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec)
|
|
435 |
apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
|
|
436 |
apply (auto simp add: diff_def add_ac)
|
|
437 |
done
|
|
438 |
|
|
439 |
text{*Product of functions - Proof is trivial but tedious
|
|
440 |
and long due to rearrangement of terms*}
|
|
441 |
|
|
442 |
lemma lemma_nsderiv1: "((a::hypreal)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
|
|
443 |
by (simp add: right_diff_distrib)
|
|
444 |
|
|
445 |
lemma lemma_nsderiv2: "[| (x - y) / z = hypreal_of_real D + yb; z \<noteq> 0;
|
|
446 |
z \<in> Infinitesimal; yb \<in> Infinitesimal |]
|
|
447 |
==> x - y \<approx> 0"
|
|
448 |
apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
|
|
449 |
apply (erule_tac V = "(x - y) / z = hypreal_of_real D + yb" in thin_rl)
|
|
450 |
apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
|
|
451 |
simp add: mult_assoc mem_infmal_iff [symmetric])
|
|
452 |
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
|
|
453 |
done
|
|
454 |
|
|
455 |
lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
|
|
456 |
==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
|
|
457 |
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
|
|
458 |
apply (auto dest!: spec
|
|
459 |
simp add: starfun_lambda_cancel lemma_nsderiv1)
|
|
460 |
apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
|
|
461 |
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
|
|
462 |
apply (auto simp add: times_divide_eq_right [symmetric]
|
|
463 |
simp del: times_divide_eq)
|
|
464 |
apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
|
|
465 |
apply (drule_tac
|
|
466 |
approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
|
|
467 |
apply (auto intro!: approx_add_mono1
|
|
468 |
simp add: left_distrib right_distrib mult_commute add_assoc)
|
|
469 |
apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
|
|
470 |
in add_commute [THEN subst])
|
|
471 |
apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
|
|
472 |
Infinitesimal_add Infinitesimal_mult
|
|
473 |
Infinitesimal_hypreal_of_real_mult
|
|
474 |
Infinitesimal_hypreal_of_real_mult2
|
|
475 |
simp add: add_assoc [symmetric])
|
|
476 |
done
|
|
477 |
|
|
478 |
text{*Multiplying by a constant*}
|
|
479 |
lemma NSDERIV_cmult: "NSDERIV f x :> D
|
|
480 |
==> NSDERIV (%x. c * f x) x :> c*D"
|
|
481 |
apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
|
|
482 |
minus_mult_right right_diff_distrib [symmetric])
|
|
483 |
apply (erule NSLIM_const [THEN NSLIM_mult])
|
|
484 |
done
|
|
485 |
|
|
486 |
text{*Negation of function*}
|
|
487 |
lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
|
|
488 |
proof (simp add: NSDERIV_NSLIM_iff)
|
|
489 |
assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D"
|
|
490 |
hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D"
|
|
491 |
by (rule NSLIM_minus)
|
|
492 |
have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
|
|
493 |
by (simp add: minus_divide_left)
|
|
494 |
with deriv
|
|
495 |
show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
|
|
496 |
qed
|
|
497 |
|
|
498 |
text{*Subtraction*}
|
|
499 |
lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
|
|
500 |
by (blast dest: NSDERIV_add NSDERIV_minus)
|
|
501 |
|
|
502 |
lemma NSDERIV_diff:
|
|
503 |
"[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
|
|
504 |
==> NSDERIV (%x. f x - g x) x :> Da-Db"
|
|
505 |
apply (simp add: diff_minus)
|
|
506 |
apply (blast intro: NSDERIV_add_minus)
|
|
507 |
done
|
|
508 |
|
|
509 |
text{* Similarly to the above, the chain rule admits an entirely
|
|
510 |
straightforward derivation. Compare this with Harrison's
|
|
511 |
HOL proof of the chain rule, which proved to be trickier and
|
|
512 |
required an alternative characterisation of differentiability-
|
|
513 |
the so-called Carathedory derivative. Our main problem is
|
|
514 |
manipulation of terms.*}
|
|
515 |
|
|
516 |
|
|
517 |
(* lemmas *)
|
|
518 |
lemma NSDERIV_zero:
|
|
519 |
"[| NSDERIV g x :> D;
|
|
520 |
( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
|
|
521 |
xa \<in> Infinitesimal;
|
|
522 |
xa \<noteq> 0
|
|
523 |
|] ==> D = 0"
|
|
524 |
apply (simp add: nsderiv_def)
|
|
525 |
apply (drule bspec, auto)
|
|
526 |
done
|
|
527 |
|
|
528 |
(* can be proved differently using NSLIM_isCont_iff *)
|
|
529 |
lemma NSDERIV_approx:
|
|
530 |
"[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
|
|
531 |
==> ( *f* f) (hypreal_of_real(x) + h) - hypreal_of_real(f x) \<approx> 0"
|
|
532 |
apply (simp add: nsderiv_def)
|
|
533 |
apply (simp add: mem_infmal_iff [symmetric])
|
|
534 |
apply (rule Infinitesimal_ratio)
|
|
535 |
apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
|
|
536 |
done
|
|
537 |
|
|
538 |
(*---------------------------------------------------------------
|
|
539 |
from one version of differentiability
|
|
540 |
|
|
541 |
f(x) - f(a)
|
|
542 |
--------------- \<approx> Db
|
|
543 |
x - a
|
|
544 |
---------------------------------------------------------------*)
|
|
545 |
lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
|
|
546 |
( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
|
|
547 |
( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
|
|
548 |
|] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
|
|
549 |
- hypreal_of_real (f (g x)))
|
|
550 |
/ (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real (g x))
|
|
551 |
\<approx> hypreal_of_real(Da)"
|
|
552 |
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
|
|
553 |
|
|
554 |
(*--------------------------------------------------------------
|
|
555 |
from other version of differentiability
|
|
556 |
|
|
557 |
f(x + h) - f(x)
|
|
558 |
----------------- \<approx> Db
|
|
559 |
h
|
|
560 |
--------------------------------------------------------------*)
|
|
561 |
lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
|
|
562 |
==> (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real(g x)) / xa
|
|
563 |
\<approx> hypreal_of_real(Db)"
|
|
564 |
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
|
|
565 |
|
|
566 |
lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
|
|
567 |
by auto
|
|
568 |
|
|
569 |
text{*This proof uses both definitions of differentiability.*}
|
|
570 |
lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
|
|
571 |
==> NSDERIV (f o g) x :> Da * Db"
|
|
572 |
apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
|
|
573 |
mem_infmal_iff [symmetric])
|
|
574 |
apply clarify
|
|
575 |
apply (frule_tac f = g in NSDERIV_approx)
|
|
576 |
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
|
|
577 |
apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
|
|
578 |
apply (drule_tac g = g in NSDERIV_zero)
|
|
579 |
apply (auto simp add: divide_inverse)
|
|
580 |
apply (rule_tac z1 = "( *f* g) (hypreal_of_real (x) + xa) - hypreal_of_real (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst])
|
|
581 |
apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
|
|
582 |
apply (rule approx_mult_hypreal_of_real)
|
|
583 |
apply (simp_all add: divide_inverse [symmetric])
|
|
584 |
apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
|
|
585 |
apply (blast intro: NSDERIVD2)
|
|
586 |
done
|
|
587 |
|
|
588 |
text{*Differentiation of natural number powers*}
|
|
589 |
lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
|
|
590 |
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
|
|
591 |
|
|
592 |
lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
|
|
593 |
by (cut_tac c = c and x = x in NSDERIV_Id [THEN NSDERIV_cmult], simp)
|
|
594 |
|
|
595 |
(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
|
|
596 |
lemma NSDERIV_inverse:
|
|
597 |
"x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
|
|
598 |
apply (simp add: nsderiv_def)
|
|
599 |
apply (rule ballI, simp, clarify)
|
|
600 |
apply (frule (1) Infinitesimal_add_not_zero)
|
|
601 |
apply (simp add: add_commute)
|
|
602 |
(*apply (auto simp add: starfun_inverse_inverse realpow_two
|
|
603 |
simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*)
|
|
604 |
apply (simp add: inverse_add inverse_mult_distrib [symmetric]
|
|
605 |
inverse_minus_eq [symmetric] add_ac mult_ac diff_def
|
|
606 |
del: inverse_mult_distrib inverse_minus_eq
|
|
607 |
minus_mult_left [symmetric] minus_mult_right [symmetric])
|
|
608 |
apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
|
|
609 |
del: minus_mult_left [symmetric] minus_mult_right [symmetric])
|
|
610 |
apply (rule_tac y = "inverse (- hypreal_of_real x * hypreal_of_real x)" in approx_trans)
|
|
611 |
apply (rule inverse_add_Infinitesimal_approx2)
|
|
612 |
apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
|
|
613 |
simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
|
|
614 |
apply (rule Infinitesimal_HFinite_mult2, auto)
|
|
615 |
done
|
|
616 |
|
|
617 |
subsubsection {* Equivalence of NS and Standard definitions *}
|
|
618 |
|
|
619 |
lemma divideR_eq_divide [simp]: "x /# y = x / y"
|
|
620 |
by (simp add: real_scaleR_def divide_inverse mult_commute)
|
|
621 |
|
|
622 |
text{*Now equivalence between NSDERIV and DERIV*}
|
|
623 |
lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
|
|
624 |
by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
|
|
625 |
|
|
626 |
(* let's do the standard proof though theorem *)
|
|
627 |
(* LIM_mult2 follows from a NS proof *)
|
|
628 |
|
|
629 |
lemma DERIV_cmult:
|
|
630 |
fixes f :: "real \<Rightarrow> 'a::real_normed_algebra" shows
|
|
631 |
"DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
|
|
632 |
by (drule DERIV_mult' [OF DERIV_const], simp)
|
|
633 |
|
|
634 |
(* standard version *)
|
|
635 |
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
|
|
636 |
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
|
|
637 |
|
|
638 |
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
|
|
639 |
by (auto dest: DERIV_chain simp add: o_def)
|
|
640 |
|
|
641 |
(*derivative of linear multiplication*)
|
|
642 |
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
|
|
643 |
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
|
|
644 |
|
|
645 |
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
|
|
646 |
apply (cut_tac DERIV_power [OF DERIV_Id])
|
|
647 |
apply (simp add: real_scaleR_def real_of_nat_def)
|
|
648 |
done
|
|
649 |
|
|
650 |
(* NS version *)
|
|
651 |
lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
|
|
652 |
by (simp add: NSDERIV_DERIV_iff DERIV_pow)
|
|
653 |
|
|
654 |
text{*Power of -1*}
|
|
655 |
|
|
656 |
lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
|
|
657 |
by (drule DERIV_inverse' [OF DERIV_Id], simp)
|
|
658 |
|
|
659 |
text{*Derivative of inverse*}
|
|
660 |
lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
|
|
661 |
==> DERIV (%x. inverse(f x)::real) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
|
|
662 |
by (drule (1) DERIV_inverse', simp add: mult_ac)
|
|
663 |
|
|
664 |
lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
|
|
665 |
==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
|
|
666 |
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
|
|
667 |
|
|
668 |
text{*Derivative of quotient*}
|
|
669 |
lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
|
|
670 |
==> DERIV (%y. f(y) / (g y) :: real) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
|
|
671 |
by (drule (2) DERIV_divide, simp add: mult_commute)
|
|
672 |
|
|
673 |
lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
|
|
674 |
==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
|
|
675 |
- (e*f(x))) / (g(x) ^ Suc (Suc 0))"
|
|
676 |
by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
|
|
677 |
|
|
678 |
lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
|
|
679 |
\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
|
|
680 |
by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV
|
|
681 |
real_scaleR_def mult_commute)
|
|
682 |
|
|
683 |
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
|
|
684 |
by auto
|
|
685 |
|
|
686 |
lemma CARAT_DERIVD:
|
|
687 |
assumes all: "\<forall>z. f z - f x = g z * (z-x)"
|
|
688 |
and nsc: "isNSCont g x"
|
|
689 |
shows "NSDERIV f x :> g x"
|
|
690 |
proof -
|
|
691 |
from nsc
|
|
692 |
have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
|
|
693 |
( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
|
|
694 |
hypreal_of_real (g x)"
|
|
695 |
by (simp add: diff_minus isNSCont_def)
|
|
696 |
thus ?thesis using all
|
|
697 |
by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
|
|
698 |
qed
|
|
699 |
|
|
700 |
subsubsection {* Differentiability predicate *}
|
|
701 |
|
|
702 |
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
|
|
703 |
by (simp add: differentiable_def)
|
|
704 |
|
|
705 |
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
|
|
706 |
by (force simp add: differentiable_def)
|
|
707 |
|
|
708 |
lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
|
|
709 |
by (simp add: NSdifferentiable_def)
|
|
710 |
|
|
711 |
lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
|
|
712 |
by (force simp add: NSdifferentiable_def)
|
|
713 |
|
|
714 |
lemma differentiable_const: "(\<lambda>z. a) differentiable x"
|
|
715 |
apply (unfold differentiable_def)
|
|
716 |
apply (rule_tac x=0 in exI)
|
|
717 |
apply simp
|
|
718 |
done
|
|
719 |
|
|
720 |
lemma differentiable_sum:
|
|
721 |
assumes "f differentiable x"
|
|
722 |
and "g differentiable x"
|
|
723 |
shows "(\<lambda>x. f x + g x) differentiable x"
|
|
724 |
proof -
|
|
725 |
from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
|
|
726 |
then obtain df where "DERIV f x :> df" ..
|
|
727 |
moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
|
|
728 |
then obtain dg where "DERIV g x :> dg" ..
|
|
729 |
ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
|
|
730 |
hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
|
|
731 |
thus ?thesis by (fold differentiable_def)
|
|
732 |
qed
|
|
733 |
|
|
734 |
lemma differentiable_diff:
|
|
735 |
assumes "f differentiable x"
|
|
736 |
and "g differentiable x"
|
|
737 |
shows "(\<lambda>x. f x - g x) differentiable x"
|
|
738 |
proof -
|
|
739 |
from prems have "f differentiable x" by simp
|
|
740 |
moreover
|
|
741 |
from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
|
|
742 |
then obtain dg where "DERIV g x :> dg" ..
|
|
743 |
then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
|
|
744 |
hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
|
|
745 |
hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
|
|
746 |
ultimately
|
|
747 |
show ?thesis
|
|
748 |
by (auto simp: real_diff_def dest: differentiable_sum)
|
|
749 |
qed
|
|
750 |
|
|
751 |
lemma differentiable_mult:
|
|
752 |
assumes "f differentiable x"
|
|
753 |
and "g differentiable x"
|
|
754 |
shows "(\<lambda>x. f x * g x) differentiable x"
|
|
755 |
proof -
|
|
756 |
from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
|
|
757 |
then obtain df where "DERIV f x :> df" ..
|
|
758 |
moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
|
|
759 |
then obtain dg where "DERIV g x :> dg" ..
|
|
760 |
ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
|
|
761 |
hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
|
|
762 |
thus ?thesis by (fold differentiable_def)
|
|
763 |
qed
|
|
764 |
|
|
765 |
subsection {*(NS) Increment*}
|
|
766 |
lemma incrementI:
|
|
767 |
"f NSdifferentiable x ==>
|
|
768 |
increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
|
|
769 |
hypreal_of_real (f x)"
|
|
770 |
by (simp add: increment_def)
|
|
771 |
|
|
772 |
lemma incrementI2: "NSDERIV f x :> D ==>
|
|
773 |
increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
|
|
774 |
hypreal_of_real (f x)"
|
|
775 |
apply (erule NSdifferentiableI [THEN incrementI])
|
|
776 |
done
|
|
777 |
|
|
778 |
(* The Increment theorem -- Keisler p. 65 *)
|
|
779 |
lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
|
|
780 |
==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
|
|
781 |
apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
|
|
782 |
apply (drule bspec, auto)
|
|
783 |
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
|
|
784 |
apply (frule_tac b1 = "hypreal_of_real (D) + y"
|
|
785 |
in hypreal_mult_right_cancel [THEN iffD2])
|
|
786 |
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)
|
|
787 |
apply assumption
|
|
788 |
apply (simp add: times_divide_eq_right [symmetric])
|
|
789 |
apply (auto simp add: left_distrib)
|
|
790 |
done
|
|
791 |
|
|
792 |
lemma increment_thm2:
|
|
793 |
"[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
|
|
794 |
==> \<exists>e \<in> Infinitesimal. increment f x h =
|
|
795 |
hypreal_of_real(D)*h + e*h"
|
|
796 |
by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
|
|
797 |
|
|
798 |
|
|
799 |
lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
|
|
800 |
==> increment f x h \<approx> 0"
|
|
801 |
apply (drule increment_thm2,
|
|
802 |
auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
|
|
803 |
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
|
|
804 |
done
|
|
805 |
|
|
806 |
subsection {* Nested Intervals and Bisection *}
|
|
807 |
|
|
808 |
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
|
|
809 |
All considerably tidied by lcp.*}
|
|
810 |
|
|
811 |
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)"
|
|
812 |
apply (induct "no")
|
|
813 |
apply (auto intro: order_trans)
|
|
814 |
done
|
|
815 |
|
|
816 |
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
|
|
817 |
\<forall>n. g(Suc n) \<le> g(n);
|
|
818 |
\<forall>n. f(n) \<le> g(n) |]
|
|
819 |
==> Bseq (f :: nat \<Rightarrow> real)"
|
|
820 |
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
|
|
821 |
apply (induct_tac "n")
|
|
822 |
apply (auto intro: order_trans)
|
|
823 |
apply (rule_tac y = "g (Suc na)" in order_trans)
|
|
824 |
apply (induct_tac [2] "na")
|
|
825 |
apply (auto intro: order_trans)
|
|
826 |
done
|
|
827 |
|
|
828 |
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
|
|
829 |
\<forall>n. g(Suc n) \<le> g(n);
|
|
830 |
\<forall>n. f(n) \<le> g(n) |]
|
|
831 |
==> Bseq (g :: nat \<Rightarrow> real)"
|
|
832 |
apply (subst Bseq_minus_iff [symmetric])
|
|
833 |
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
|
|
834 |
apply auto
|
|
835 |
done
|
|
836 |
|
|
837 |
lemma f_inc_imp_le_lim:
|
|
838 |
fixes f :: "nat \<Rightarrow> real"
|
|
839 |
shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
|
|
840 |
apply (rule linorder_not_less [THEN iffD1])
|
|
841 |
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
|
|
842 |
apply (drule real_less_sum_gt_zero)
|
|
843 |
apply (drule_tac x = "f n + - lim f" in spec, safe)
|
|
844 |
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
|
|
845 |
apply (subgoal_tac "lim f \<le> f (no + n) ")
|
|
846 |
apply (drule_tac no=no and m=n in lemma_f_mono_add)
|
|
847 |
apply (auto simp add: add_commute)
|
|
848 |
apply (induct_tac "no")
|
|
849 |
apply simp
|
|
850 |
apply (auto intro: order_trans simp add: diff_minus abs_if)
|
|
851 |
done
|
|
852 |
|
|
853 |
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
|
|
854 |
apply (rule LIMSEQ_minus [THEN limI])
|
|
855 |
apply (simp add: convergent_LIMSEQ_iff)
|
|
856 |
done
|
|
857 |
|
|
858 |
lemma g_dec_imp_lim_le:
|
|
859 |
fixes g :: "nat \<Rightarrow> real"
|
|
860 |
shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
|
|
861 |
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
|
|
862 |
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
|
|
863 |
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
|
|
864 |
done
|
|
865 |
|
|
866 |
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
|
|
867 |
\<forall>n. g(Suc n) \<le> g(n);
|
|
868 |
\<forall>n. f(n) \<le> g(n) |]
|
|
869 |
==> \<exists>l m :: real. l \<le> m & ((\<forall>n. f(n) \<le> l) & f ----> l) &
|
|
870 |
((\<forall>n. m \<le> g(n)) & g ----> m)"
|
|
871 |
apply (subgoal_tac "monoseq f & monoseq g")
|
|
872 |
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
|
|
873 |
apply (subgoal_tac "Bseq f & Bseq g")
|
|
874 |
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
|
|
875 |
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
|
|
876 |
apply (rule_tac x = "lim f" in exI)
|
|
877 |
apply (rule_tac x = "lim g" in exI)
|
|
878 |
apply (auto intro: LIMSEQ_le)
|
|
879 |
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
|
|
880 |
done
|
|
881 |
|
|
882 |
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
|
|
883 |
\<forall>n. g(Suc n) \<le> g(n);
|
|
884 |
\<forall>n. f(n) \<le> g(n);
|
|
885 |
(%n. f(n) - g(n)) ----> 0 |]
|
|
886 |
==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
|
|
887 |
((\<forall>n. l \<le> g(n)) & g ----> l)"
|
|
888 |
apply (drule lemma_nest, auto)
|
|
889 |
apply (subgoal_tac "l = m")
|
|
890 |
apply (drule_tac [2] X = f in LIMSEQ_diff)
|
|
891 |
apply (auto intro: LIMSEQ_unique)
|
|
892 |
done
|
|
893 |
|
|
894 |
text{*The universal quantifiers below are required for the declaration
|
|
895 |
of @{text Bolzano_nest_unique} below.*}
|
|
896 |
|
|
897 |
lemma Bolzano_bisect_le:
|
|
898 |
"a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
|
|
899 |
apply (rule allI)
|
|
900 |
apply (induct_tac "n")
|
|
901 |
apply (auto simp add: Let_def split_def)
|
|
902 |
done
|
|
903 |
|
|
904 |
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
|
|
905 |
\<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
|
|
906 |
apply (rule allI)
|
|
907 |
apply (induct_tac "n")
|
|
908 |
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
|
|
909 |
done
|
|
910 |
|
|
911 |
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
|
|
912 |
\<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
|
|
913 |
apply (rule allI)
|
|
914 |
apply (induct_tac "n")
|
|
915 |
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
|
|
916 |
done
|
|
917 |
|
|
918 |
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
|
|
919 |
apply (auto)
|
|
920 |
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
|
|
921 |
apply (simp)
|
|
922 |
done
|
|
923 |
|
|
924 |
lemma Bolzano_bisect_diff:
|
|
925 |
"a \<le> b ==>
|
|
926 |
snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
|
|
927 |
(b-a) / (2 ^ n)"
|
|
928 |
apply (induct "n")
|
|
929 |
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
|
|
930 |
done
|
|
931 |
|
|
932 |
lemmas Bolzano_nest_unique =
|
|
933 |
lemma_nest_unique
|
|
934 |
[OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
|
|
935 |
|
|
936 |
|
|
937 |
lemma not_P_Bolzano_bisect:
|
|
938 |
assumes P: "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
|
|
939 |
and notP: "~ P(a,b)"
|
|
940 |
and le: "a \<le> b"
|
|
941 |
shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
|
|
942 |
proof (induct n)
|
|
943 |
case 0 thus ?case by simp
|
|
944 |
next
|
|
945 |
case (Suc n)
|
|
946 |
thus ?case
|
|
947 |
by (auto simp del: surjective_pairing [symmetric]
|
|
948 |
simp add: Let_def split_def Bolzano_bisect_le [OF le]
|
|
949 |
P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
|
|
950 |
qed
|
|
951 |
|
|
952 |
(*Now we re-package P_prem as a formula*)
|
|
953 |
lemma not_P_Bolzano_bisect':
|
|
954 |
"[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
|
|
955 |
~ P(a,b); a \<le> b |] ==>
|
|
956 |
\<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
|
|
957 |
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
|
|
958 |
|
|
959 |
|
|
960 |
|
|
961 |
lemma lemma_BOLZANO:
|
|
962 |
"[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
|
|
963 |
\<forall>x. \<exists>d::real. 0 < d &
|
|
964 |
(\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
|
|
965 |
a \<le> b |]
|
|
966 |
==> P(a,b)"
|
|
967 |
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
|
|
968 |
apply (rule LIMSEQ_minus_cancel)
|
|
969 |
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
|
|
970 |
apply (rule ccontr)
|
|
971 |
apply (drule not_P_Bolzano_bisect', assumption+)
|
|
972 |
apply (rename_tac "l")
|
|
973 |
apply (drule_tac x = l in spec, clarify)
|
|
974 |
apply (simp add: LIMSEQ_def)
|
|
975 |
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
|
|
976 |
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
|
|
977 |
apply (drule real_less_half_sum, auto)
|
|
978 |
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
|
|
979 |
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
|
|
980 |
apply safe
|
|
981 |
apply (simp_all (no_asm_simp))
|
|
982 |
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)
|
|
983 |
apply (simp (no_asm_simp) add: abs_if)
|
|
984 |
apply (rule real_sum_of_halves [THEN subst])
|
|
985 |
apply (rule add_strict_mono)
|
|
986 |
apply (simp_all add: diff_minus [symmetric])
|
|
987 |
done
|
|
988 |
|
|
989 |
|
|
990 |
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
|
|
991 |
(\<forall>x. \<exists>d::real. 0 < d &
|
|
992 |
(\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
|
|
993 |
--> (\<forall>a b. a \<le> b --> P(a,b))"
|
|
994 |
apply clarify
|
|
995 |
apply (blast intro: lemma_BOLZANO)
|
|
996 |
done
|
|
997 |
|
|
998 |
|
|
999 |
subsection {* Intermediate Value Theorem *}
|
|
1000 |
|
|
1001 |
text {*Prove Contrapositive by Bisection*}
|
|
1002 |
|
|
1003 |
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
|
|
1004 |
a \<le> b;
|
|
1005 |
(\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
|
|
1006 |
==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
|
|
1007 |
apply (rule contrapos_pp, assumption)
|
|
1008 |
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)
|
|
1009 |
apply safe
|
|
1010 |
apply simp_all
|
|
1011 |
apply (simp add: isCont_iff LIM_def)
|
|
1012 |
apply (rule ccontr)
|
|
1013 |
apply (subgoal_tac "a \<le> x & x \<le> b")
|
|
1014 |
prefer 2
|
|
1015 |
apply simp
|
|
1016 |
apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
|
|
1017 |
apply (drule_tac x = x in spec)+
|
|
1018 |
apply simp
|
|
1019 |
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
|
|
1020 |
apply safe
|
|
1021 |
apply simp
|
|
1022 |
apply (drule_tac x = s in spec, clarify)
|
|
1023 |
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
|
|
1024 |
apply (drule_tac x = "ba-x" in spec)
|
|
1025 |
apply (simp_all add: abs_if)
|
|
1026 |
apply (drule_tac x = "aa-x" in spec)
|
|
1027 |
apply (case_tac "x \<le> aa", simp_all)
|
|
1028 |
done
|
|
1029 |
|
|
1030 |
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
|
|
1031 |
a \<le> b;
|
|
1032 |
(\<forall>x. a \<le> x & x \<le> b --> isCont f x)
|
|
1033 |
|] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
|
|
1034 |
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
|
|
1035 |
apply (drule IVT [where f = "%x. - f x"], assumption)
|
|
1036 |
apply (auto intro: isCont_minus)
|
|
1037 |
done
|
|
1038 |
|
|
1039 |
(*HOL style here: object-level formulations*)
|
|
1040 |
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
|
|
1041 |
(\<forall>x. a \<le> x & x \<le> b --> isCont f x))
|
|
1042 |
--> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
|
|
1043 |
apply (blast intro: IVT)
|
|
1044 |
done
|
|
1045 |
|
|
1046 |
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
|
|
1047 |
(\<forall>x. a \<le> x & x \<le> b --> isCont f x))
|
|
1048 |
--> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
|
|
1049 |
apply (blast intro: IVT2)
|
|
1050 |
done
|
|
1051 |
|
|
1052 |
text{*By bisection, function continuous on closed interval is bounded above*}
|
|
1053 |
|
|
1054 |
lemma isCont_bounded:
|
|
1055 |
"[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
|
|
1056 |
==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
|
|
1057 |
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)
|
|
1058 |
apply safe
|
|
1059 |
apply simp_all
|
|
1060 |
apply (rename_tac x xa ya M Ma)
|
|
1061 |
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
|
|
1062 |
apply (rule_tac x = Ma in exI, clarify)
|
|
1063 |
apply (cut_tac x = xb and y = xa in linorder_linear, force)
|
|
1064 |
apply (rule_tac x = M in exI, clarify)
|
|
1065 |
apply (cut_tac x = xb and y = xa in linorder_linear, force)
|
|
1066 |
apply (case_tac "a \<le> x & x \<le> b")
|
|
1067 |
apply (rule_tac [2] x = 1 in exI)
|
|
1068 |
prefer 2 apply force
|
|
1069 |
apply (simp add: LIM_def isCont_iff)
|
|
1070 |
apply (drule_tac x = x in spec, auto)
|
|
1071 |
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
|
|
1072 |
apply (drule_tac x = 1 in spec, auto)
|
|
1073 |
apply (rule_tac x = s in exI, clarify)
|
|
1074 |
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
|
|
1075 |
apply (drule_tac x = "xa-x" in spec)
|
|
1076 |
apply (auto simp add: abs_ge_self)
|
|
1077 |
done
|
|
1078 |
|
|
1079 |
text{*Refine the above to existence of least upper bound*}
|
|
1080 |
|
|
1081 |
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
|
|
1082 |
(\<exists>t. isLub UNIV S t)"
|
|
1083 |
by (blast intro: reals_complete)
|
|
1084 |
|
|
1085 |
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
|
|
1086 |
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
|
|
1087 |
(\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
|
|
1088 |
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
|
|
1089 |
in lemma_reals_complete)
|
|
1090 |
apply auto
|
|
1091 |
apply (drule isCont_bounded, assumption)
|
|
1092 |
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
|
|
1093 |
apply (rule exI, auto)
|
|
1094 |
apply (auto dest!: spec simp add: linorder_not_less)
|
|
1095 |
done
|
|
1096 |
|
|
1097 |
text{*Now show that it attains its upper bound*}
|
|
1098 |
|
|
1099 |
lemma isCont_eq_Ub:
|
|
1100 |
assumes le: "a \<le> b"
|
|
1101 |
and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
|
|
1102 |
shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
|
|
1103 |
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
|
|
1104 |
proof -
|
|
1105 |
from isCont_has_Ub [OF le con]
|
|
1106 |
obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
|
|
1107 |
and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x" by blast
|
|
1108 |
show ?thesis
|
|
1109 |
proof (intro exI, intro conjI)
|
|
1110 |
show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
|
|
1111 |
show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
|
|
1112 |
proof (rule ccontr)
|
|
1113 |
assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
|
|
1114 |
with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
|
|
1115 |
by (fastsimp simp add: linorder_not_le [symmetric])
|
|
1116 |
hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
|
|
1117 |
by (auto simp add: isCont_inverse isCont_diff con)
|
|
1118 |
from isCont_bounded [OF le this]
|
|
1119 |
obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
|
|
1120 |
have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
|
|
1121 |
by (simp add: M3 compare_rls)
|
|
1122 |
have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
|
|
1123 |
by (auto intro: order_le_less_trans [of _ k])
|
|
1124 |
with Minv
|
|
1125 |
have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
|
|
1126 |
by (intro strip less_imp_inverse_less, simp_all)
|
|
1127 |
hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
|
|
1128 |
by simp
|
|
1129 |
have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
|
|
1130 |
by (simp, arith)
|
|
1131 |
from M2 [OF this]
|
|
1132 |
obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
|
|
1133 |
thus False using invlt [of x] by force
|
|
1134 |
qed
|
|
1135 |
qed
|
|
1136 |
qed
|
|
1137 |
|
|
1138 |
|
|
1139 |
text{*Same theorem for lower bound*}
|
|
1140 |
|
|
1141 |
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
|
|
1142 |
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
|
|
1143 |
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
|
|
1144 |
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
|
|
1145 |
prefer 2 apply (blast intro: isCont_minus)
|
|
1146 |
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
|
|
1147 |
apply safe
|
|
1148 |
apply auto
|
|
1149 |
done
|
|
1150 |
|
|
1151 |
|
|
1152 |
text{*Another version.*}
|
|
1153 |
|
|
1154 |
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
|
|
1155 |
==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
|
|
1156 |
(\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
|
|
1157 |
apply (frule isCont_eq_Lb)
|
|
1158 |
apply (frule_tac [2] isCont_eq_Ub)
|
|
1159 |
apply (assumption+, safe)
|
|
1160 |
apply (rule_tac x = "f x" in exI)
|
|
1161 |
apply (rule_tac x = "f xa" in exI, simp, safe)
|
|
1162 |
apply (cut_tac x = x and y = xa in linorder_linear, safe)
|
|
1163 |
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
|
|
1164 |
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
|
|
1165 |
apply (rule_tac [2] x = xb in exI)
|
|
1166 |
apply (rule_tac [4] x = xb in exI, simp_all)
|
|
1167 |
done
|
|
1168 |
|
|
1169 |
|
|
1170 |
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
|
|
1171 |
|
|
1172 |
lemma DERIV_left_inc:
|
|
1173 |
fixes f :: "real => real"
|
|
1174 |
assumes der: "DERIV f x :> l"
|
|
1175 |
and l: "0 < l"
|
|
1176 |
shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
|
|
1177 |
proof -
|
|
1178 |
from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
|
|
1179 |
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)"
|
|
1180 |
by (simp add: diff_minus)
|
|
1181 |
then obtain s
|
|
1182 |
where s: "0 < s"
|
|
1183 |
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
|
|
1184 |
by auto
|
|
1185 |
thus ?thesis
|
|
1186 |
proof (intro exI conjI strip)
|
|
1187 |
show "0<s" .
|
|
1188 |
fix h::real
|
|
1189 |
assume "0 < h" "h < s"
|
|
1190 |
with all [of h] show "f x < f (x+h)"
|
|
1191 |
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
|
|
1192 |
split add: split_if_asm)
|
|
1193 |
assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
|
|
1194 |
with l
|
|
1195 |
have "0 < (f (x+h) - f x) / h" by arith
|
|
1196 |
thus "f x < f (x+h)"
|
|
1197 |
by (simp add: pos_less_divide_eq h)
|
|
1198 |
qed
|
|
1199 |
qed
|
|
1200 |
qed
|
|
1201 |
|
|
1202 |
lemma DERIV_left_dec:
|
|
1203 |
fixes f :: "real => real"
|
|
1204 |
assumes der: "DERIV f x :> l"
|
|
1205 |
and l: "l < 0"
|
|
1206 |
shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
|
|
1207 |
proof -
|
|
1208 |
from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
|
|
1209 |
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)"
|
|
1210 |
by (simp add: diff_minus)
|
|
1211 |
then obtain s
|
|
1212 |
where s: "0 < s"
|
|
1213 |
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
|
|
1214 |
by auto
|
|
1215 |
thus ?thesis
|
|
1216 |
proof (intro exI conjI strip)
|
|
1217 |
show "0<s" .
|
|
1218 |
fix h::real
|
|
1219 |
assume "0 < h" "h < s"
|
|
1220 |
with all [of "-h"] show "f x < f (x-h)"
|
|
1221 |
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
|
|
1222 |
split add: split_if_asm)
|
|
1223 |
assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
|
|
1224 |
with l
|
|
1225 |
have "0 < (f (x-h) - f x) / h" by arith
|
|
1226 |
thus "f x < f (x-h)"
|
|
1227 |
by (simp add: pos_less_divide_eq h)
|
|
1228 |
qed
|
|
1229 |
qed
|
|
1230 |
qed
|
|
1231 |
|
|
1232 |
lemma DERIV_local_max:
|
|
1233 |
fixes f :: "real => real"
|
|
1234 |
assumes der: "DERIV f x :> l"
|
|
1235 |
and d: "0 < d"
|
|
1236 |
and le: "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
|
|
1237 |
shows "l = 0"
|
|
1238 |
proof (cases rule: linorder_cases [of l 0])
|
|
1239 |
case equal show ?thesis .
|
|
1240 |
next
|
|
1241 |
case less
|
|
1242 |
from DERIV_left_dec [OF der less]
|
|
1243 |
obtain d' where d': "0 < d'"
|
|
1244 |
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
|
|
1245 |
from real_lbound_gt_zero [OF d d']
|
|
1246 |
obtain e where "0 < e \<and> e < d \<and> e < d'" ..
|
|
1247 |
with lt le [THEN spec [where x="x-e"]]
|
|
1248 |
show ?thesis by (auto simp add: abs_if)
|
|
1249 |
next
|
|
1250 |
case greater
|
|
1251 |
from DERIV_left_inc [OF der greater]
|
|
1252 |
obtain d' where d': "0 < d'"
|
|
1253 |
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
|
|
1254 |
from real_lbound_gt_zero [OF d d']
|
|
1255 |
obtain e where "0 < e \<and> e < d \<and> e < d'" ..
|
|
1256 |
with lt le [THEN spec [where x="x+e"]]
|
|
1257 |
show ?thesis by (auto simp add: abs_if)
|
|
1258 |
qed
|
|
1259 |
|
|
1260 |
|
|
1261 |
text{*Similar theorem for a local minimum*}
|
|
1262 |
lemma DERIV_local_min:
|
|
1263 |
fixes f :: "real => real"
|
|
1264 |
shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
|
|
1265 |
by (drule DERIV_minus [THEN DERIV_local_max], auto)
|
|
1266 |
|
|
1267 |
|
|
1268 |
text{*In particular, if a function is locally flat*}
|
|
1269 |
lemma DERIV_local_const:
|
|
1270 |
fixes f :: "real => real"
|
|
1271 |
shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
|
|
1272 |
by (auto dest!: DERIV_local_max)
|
|
1273 |
|
|
1274 |
text{*Lemma about introducing open ball in open interval*}
|
|
1275 |
lemma lemma_interval_lt:
|
|
1276 |
"[| a < x; x < b |]
|
|
1277 |
==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
|
|
1278 |
apply (simp add: abs_interval_iff)
|
|
1279 |
apply (insert linorder_linear [of "x-a" "b-x"], safe)
|
|
1280 |
apply (rule_tac x = "x-a" in exI)
|
|
1281 |
apply (rule_tac [2] x = "b-x" in exI, auto)
|
|
1282 |
done
|
|
1283 |
|
|
1284 |
lemma lemma_interval: "[| a < x; x < b |] ==>
|
|
1285 |
\<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
|
|
1286 |
apply (drule lemma_interval_lt, auto)
|
|
1287 |
apply (auto intro!: exI)
|
|
1288 |
done
|
|
1289 |
|
|
1290 |
text{*Rolle's Theorem.
|
|
1291 |
If @{term f} is defined and continuous on the closed interval
|
|
1292 |
@{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
|
|
1293 |
and @{term "f(a) = f(b)"},
|
|
1294 |
then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
|
|
1295 |
theorem Rolle:
|
|
1296 |
assumes lt: "a < b"
|
|
1297 |
and eq: "f(a) = f(b)"
|
|
1298 |
and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
|
|
1299 |
and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
|
|
1300 |
shows "\<exists>z. a < z & z < b & DERIV f z :> 0"
|
|
1301 |
proof -
|
|
1302 |
have le: "a \<le> b" using lt by simp
|
|
1303 |
from isCont_eq_Ub [OF le con]
|
|
1304 |
obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
|
|
1305 |
and alex: "a \<le> x" and xleb: "x \<le> b"
|
|
1306 |
by blast
|
|
1307 |
from isCont_eq_Lb [OF le con]
|
|
1308 |
obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
|
|
1309 |
and alex': "a \<le> x'" and x'leb: "x' \<le> b"
|
|
1310 |
by blast
|
|
1311 |
show ?thesis
|
|
1312 |
proof cases
|
|
1313 |
assume axb: "a < x & x < b"
|
|
1314 |
--{*@{term f} attains its maximum within the interval*}
|
|
1315 |
hence ax: "a<x" and xb: "x<b" by auto
|
|
1316 |
from lemma_interval [OF ax xb]
|
|
1317 |
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
|
|
1318 |
by blast
|
|
1319 |
hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
|
|
1320 |
by blast
|
|
1321 |
from differentiableD [OF dif [OF axb]]
|
|
1322 |
obtain l where der: "DERIV f x :> l" ..
|
|
1323 |
have "l=0" by (rule DERIV_local_max [OF der d bound'])
|
|
1324 |
--{*the derivative at a local maximum is zero*}
|
|
1325 |
thus ?thesis using ax xb der by auto
|
|
1326 |
next
|
|
1327 |
assume notaxb: "~ (a < x & x < b)"
|
|
1328 |
hence xeqab: "x=a | x=b" using alex xleb by arith
|
|
1329 |
hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
|
|
1330 |
show ?thesis
|
|
1331 |
proof cases
|
|
1332 |
assume ax'b: "a < x' & x' < b"
|
|
1333 |
--{*@{term f} attains its minimum within the interval*}
|
|
1334 |
hence ax': "a<x'" and x'b: "x'<b" by auto
|
|
1335 |
from lemma_interval [OF ax' x'b]
|
|
1336 |
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
|
|
1337 |
by blast
|
|
1338 |
hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
|
|
1339 |
by blast
|
|
1340 |
from differentiableD [OF dif [OF ax'b]]
|
|
1341 |
obtain l where der: "DERIV f x' :> l" ..
|
|
1342 |
have "l=0" by (rule DERIV_local_min [OF der d bound'])
|
|
1343 |
--{*the derivative at a local minimum is zero*}
|
|
1344 |
thus ?thesis using ax' x'b der by auto
|
|
1345 |
next
|
|
1346 |
assume notax'b: "~ (a < x' & x' < b)"
|
|
1347 |
--{*@{term f} is constant througout the interval*}
|
|
1348 |
hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
|
|
1349 |
hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
|
|
1350 |
from dense [OF lt]
|
|
1351 |
obtain r where ar: "a < r" and rb: "r < b" by blast
|
|
1352 |
from lemma_interval [OF ar rb]
|
|
1353 |
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
|
|
1354 |
by blast
|
|
1355 |
have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
|
|
1356 |
proof (clarify)
|
|
1357 |
fix z::real
|
|
1358 |
assume az: "a \<le> z" and zb: "z \<le> b"
|
|
1359 |
show "f z = f b"
|
|
1360 |
proof (rule order_antisym)
|
|
1361 |
show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
|
|
1362 |
show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
|
|
1363 |
qed
|
|
1364 |
qed
|
|
1365 |
have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
|
|
1366 |
proof (intro strip)
|
|
1367 |
fix y::real
|
|
1368 |
assume lt: "\<bar>r-y\<bar> < d"
|
|
1369 |
hence "f y = f b" by (simp add: eq_fb bound)
|
|
1370 |
thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
|
|
1371 |
qed
|
|
1372 |
from differentiableD [OF dif [OF conjI [OF ar rb]]]
|
|
1373 |
obtain l where der: "DERIV f r :> l" ..
|
|
1374 |
have "l=0" by (rule DERIV_local_const [OF der d bound'])
|
|
1375 |
--{*the derivative of a constant function is zero*}
|
|
1376 |
thus ?thesis using ar rb der by auto
|
|
1377 |
qed
|
|
1378 |
qed
|
|
1379 |
qed
|
|
1380 |
|
|
1381 |
|
|
1382 |
subsection{*Mean Value Theorem*}
|
|
1383 |
|
|
1384 |
lemma lemma_MVT:
|
|
1385 |
"f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
|
|
1386 |
proof cases
|
|
1387 |
assume "a=b" thus ?thesis by simp
|
|
1388 |
next
|
|
1389 |
assume "a\<noteq>b"
|
|
1390 |
hence ba: "b-a \<noteq> 0" by arith
|
|
1391 |
show ?thesis
|
|
1392 |
by (rule real_mult_left_cancel [OF ba, THEN iffD1],
|
|
1393 |
simp add: right_diff_distrib,
|
|
1394 |
simp add: left_diff_distrib)
|
|
1395 |
qed
|
|
1396 |
|
|
1397 |
theorem MVT:
|
|
1398 |
assumes lt: "a < b"
|
|
1399 |
and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
|
|
1400 |
and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
|
|
1401 |
shows "\<exists>l z. a < z & z < b & DERIV f z :> l &
|
|
1402 |
(f(b) - f(a) = (b-a) * l)"
|
|
1403 |
proof -
|
|
1404 |
let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
|
|
1405 |
have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
|
|
1406 |
by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id)
|
|
1407 |
have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
|
|
1408 |
proof (clarify)
|
|
1409 |
fix x::real
|
|
1410 |
assume ax: "a < x" and xb: "x < b"
|
|
1411 |
from differentiableD [OF dif [OF conjI [OF ax xb]]]
|
|
1412 |
obtain l where der: "DERIV f x :> l" ..
|
|
1413 |
show "?F differentiable x"
|
|
1414 |
by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
|
|
1415 |
blast intro: DERIV_diff DERIV_cmult_Id der)
|
|
1416 |
qed
|
|
1417 |
from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
|
|
1418 |
obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
|
|
1419 |
by blast
|
|
1420 |
have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
|
|
1421 |
by (rule DERIV_cmult_Id)
|
|
1422 |
hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
|
|
1423 |
:> 0 + (f b - f a) / (b - a)"
|
|
1424 |
by (rule DERIV_add [OF der])
|
|
1425 |
show ?thesis
|
|
1426 |
proof (intro exI conjI)
|
|
1427 |
show "a < z" .
|
|
1428 |
show "z < b" .
|
|
1429 |
show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
|
|
1430 |
show "DERIV f z :> ((f b - f a)/(b-a))" using derF by simp
|
|
1431 |
qed
|
|
1432 |
qed
|
|
1433 |
|
|
1434 |
|
|
1435 |
text{*A function is constant if its derivative is 0 over an interval.*}
|
|
1436 |
|
|
1437 |
lemma DERIV_isconst_end:
|
|
1438 |
fixes f :: "real => real"
|
|
1439 |
shows "[| a < b;
|
|
1440 |
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
|
|
1441 |
\<forall>x. a < x & x < b --> DERIV f x :> 0 |]
|
|
1442 |
==> f b = f a"
|
|
1443 |
apply (drule MVT, assumption)
|
|
1444 |
apply (blast intro: differentiableI)
|
|
1445 |
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
|
|
1446 |
done
|
|
1447 |
|
|
1448 |
lemma DERIV_isconst1:
|
|
1449 |
fixes f :: "real => real"
|
|
1450 |
shows "[| a < b;
|
|
1451 |
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
|
|
1452 |
\<forall>x. a < x & x < b --> DERIV f x :> 0 |]
|
|
1453 |
==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
|
|
1454 |
apply safe
|
|
1455 |
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
|
|
1456 |
apply (drule_tac b = x in DERIV_isconst_end, auto)
|
|
1457 |
done
|
|
1458 |
|
|
1459 |
lemma DERIV_isconst2:
|
|
1460 |
fixes f :: "real => real"
|
|
1461 |
shows "[| a < b;
|
|
1462 |
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
|
|
1463 |
\<forall>x. a < x & x < b --> DERIV f x :> 0;
|
|
1464 |
a \<le> x; x \<le> b |]
|
|
1465 |
==> f x = f a"
|
|
1466 |
apply (blast dest: DERIV_isconst1)
|
|
1467 |
done
|
|
1468 |
|
|
1469 |
lemma DERIV_isconst_all:
|
|
1470 |
fixes f :: "real => real"
|
|
1471 |
shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
|
|
1472 |
apply (rule linorder_cases [of x y])
|
|
1473 |
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
|
|
1474 |
done
|
|
1475 |
|
|
1476 |
lemma DERIV_const_ratio_const:
|
|
1477 |
"[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
|
|
1478 |
apply (rule linorder_cases [of a b], auto)
|
|
1479 |
apply (drule_tac [!] f = f in MVT)
|
|
1480 |
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
|
|
1481 |
apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
|
|
1482 |
done
|
|
1483 |
|
|
1484 |
lemma DERIV_const_ratio_const2:
|
|
1485 |
"[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
|
|
1486 |
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
|
|
1487 |
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
|
|
1488 |
done
|
|
1489 |
|
|
1490 |
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
|
|
1491 |
by (simp)
|
|
1492 |
|
|
1493 |
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
|
|
1494 |
by (simp)
|
|
1495 |
|
|
1496 |
text{*Gallileo's "trick": average velocity = av. of end velocities*}
|
|
1497 |
|
|
1498 |
lemma DERIV_const_average:
|
|
1499 |
fixes v :: "real => real"
|
|
1500 |
assumes neq: "a \<noteq> (b::real)"
|
|
1501 |
and der: "\<forall>x. DERIV v x :> k"
|
|
1502 |
shows "v ((a + b)/2) = (v a + v b)/2"
|
|
1503 |
proof (cases rule: linorder_cases [of a b])
|
|
1504 |
case equal with neq show ?thesis by simp
|
|
1505 |
next
|
|
1506 |
case less
|
|
1507 |
have "(v b - v a) / (b - a) = k"
|
|
1508 |
by (rule DERIV_const_ratio_const2 [OF neq der])
|
|
1509 |
hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
|
|
1510 |
moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
|
|
1511 |
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
|
|
1512 |
ultimately show ?thesis using neq by force
|
|
1513 |
next
|
|
1514 |
case greater
|
|
1515 |
have "(v b - v a) / (b - a) = k"
|
|
1516 |
by (rule DERIV_const_ratio_const2 [OF neq der])
|
|
1517 |
hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
|
|
1518 |
moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
|
|
1519 |
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
|
|
1520 |
ultimately show ?thesis using neq by (force simp add: add_commute)
|
|
1521 |
qed
|
|
1522 |
|
|
1523 |
|
|
1524 |
text{*Dull lemma: an continuous injection on an interval must have a
|
|
1525 |
strict maximum at an end point, not in the middle.*}
|
|
1526 |
|
|
1527 |
lemma lemma_isCont_inj:
|
|
1528 |
fixes f :: "real \<Rightarrow> real"
|
|
1529 |
assumes d: "0 < d"
|
|
1530 |
and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
|
|
1531 |
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
|
|
1532 |
shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
|
|
1533 |
proof (rule ccontr)
|
|
1534 |
assume "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
|
|
1535 |
hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
|
|
1536 |
show False
|
|
1537 |
proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
|
|
1538 |
case le
|
|
1539 |
from d cont all [of "x+d"]
|
|
1540 |
have flef: "f(x+d) \<le> f x"
|
|
1541 |
and xlex: "x - d \<le> x"
|
|
1542 |
and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
|
|
1543 |
by (auto simp add: abs_if)
|
|
1544 |
from IVT [OF le flef xlex cont']
|
|
1545 |
obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
|
|
1546 |
moreover
|
|
1547 |
hence "g(f x') = g (f(x+d))" by simp
|
|
1548 |
ultimately show False using d inj [of x'] inj [of "x+d"]
|
|
1549 |
by (simp add: abs_le_interval_iff)
|
|
1550 |
next
|
|
1551 |
case ge
|
|
1552 |
from d cont all [of "x-d"]
|
|
1553 |
have flef: "f(x-d) \<le> f x"
|
|
1554 |
and xlex: "x \<le> x+d"
|
|
1555 |
and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
|
|
1556 |
by (auto simp add: abs_if)
|
|
1557 |
from IVT2 [OF ge flef xlex cont']
|
|
1558 |
obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
|
|
1559 |
moreover
|
|
1560 |
hence "g(f x') = g (f(x-d))" by simp
|
|
1561 |
ultimately show False using d inj [of x'] inj [of "x-d"]
|
|
1562 |
by (simp add: abs_le_interval_iff)
|
|
1563 |
qed
|
|
1564 |
qed
|
|
1565 |
|
|
1566 |
|
|
1567 |
text{*Similar version for lower bound.*}
|
|
1568 |
|
|
1569 |
lemma lemma_isCont_inj2:
|
|
1570 |
fixes f g :: "real \<Rightarrow> real"
|
|
1571 |
shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
|
|
1572 |
\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
|
|
1573 |
==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
|
|
1574 |
apply (insert lemma_isCont_inj
|
|
1575 |
[where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
|
|
1576 |
apply (simp add: isCont_minus linorder_not_le)
|
|
1577 |
done
|
|
1578 |
|
|
1579 |
text{*Show there's an interval surrounding @{term "f(x)"} in
|
|
1580 |
@{text "f[[x - d, x + d]]"} .*}
|
|
1581 |
|
|
1582 |
lemma isCont_inj_range:
|
|
1583 |
fixes f :: "real \<Rightarrow> real"
|
|
1584 |
assumes d: "0 < d"
|
|
1585 |
and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
|
|
1586 |
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
|
|
1587 |
shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
|
|
1588 |
proof -
|
|
1589 |
have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
|
|
1590 |
by (auto simp add: abs_le_interval_iff)
|
|
1591 |
from isCont_Lb_Ub [OF this]
|
|
1592 |
obtain L M
|
|
1593 |
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"
|
|
1594 |
and all2 [rule_format]:
|
|
1595 |
"\<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)"
|
|
1596 |
by auto
|
|
1597 |
with d have "L \<le> f x & f x \<le> M" by simp
|
|
1598 |
moreover have "L \<noteq> f x"
|
|
1599 |
proof -
|
|
1600 |
from lemma_isCont_inj2 [OF d inj cont]
|
|
1601 |
obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x" by auto
|
|
1602 |
thus ?thesis using all1 [of u] by arith
|
|
1603 |
qed
|
|
1604 |
moreover have "f x \<noteq> M"
|
|
1605 |
proof -
|
|
1606 |
from lemma_isCont_inj [OF d inj cont]
|
|
1607 |
obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u" by auto
|
|
1608 |
thus ?thesis using all1 [of u] by arith
|
|
1609 |
qed
|
|
1610 |
ultimately have "L < f x & f x < M" by arith
|
|
1611 |
hence "0 < f x - L" "0 < M - f x" by arith+
|
|
1612 |
from real_lbound_gt_zero [OF this]
|
|
1613 |
obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
|
|
1614 |
thus ?thesis
|
|
1615 |
proof (intro exI conjI)
|
|
1616 |
show "0<e" .
|
|
1617 |
show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
|
|
1618 |
proof (intro strip)
|
|
1619 |
fix y::real
|
|
1620 |
assume "\<bar>y - f x\<bar> \<le> e"
|
|
1621 |
with e have "L \<le> y \<and> y \<le> M" by arith
|
|
1622 |
from all2 [OF this]
|
|
1623 |
obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
|
|
1624 |
thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
|
|
1625 |
by (force simp add: abs_le_interval_iff)
|
|
1626 |
qed
|
|
1627 |
qed
|
|
1628 |
qed
|
|
1629 |
|
|
1630 |
|
|
1631 |
text{*Continuity of inverse function*}
|
|
1632 |
|
|
1633 |
lemma isCont_inverse_function:
|
|
1634 |
fixes f g :: "real \<Rightarrow> real"
|
|
1635 |
assumes d: "0 < d"
|
|
1636 |
and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
|
|
1637 |
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
|
|
1638 |
shows "isCont g (f x)"
|
|
1639 |
proof (simp add: isCont_iff LIM_eq)
|
|
1640 |
show "\<forall>r. 0 < r \<longrightarrow>
|
|
1641 |
(\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
|
|
1642 |
proof (intro strip)
|
|
1643 |
fix r::real
|
|
1644 |
assume r: "0<r"
|
|
1645 |
from real_lbound_gt_zero [OF r d]
|
|
1646 |
obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
|
|
1647 |
with inj cont
|
|
1648 |
have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
|
|
1649 |
"\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z" by auto
|
|
1650 |
from isCont_inj_range [OF e this]
|
|
1651 |
obtain e' where e': "0 < e'"
|
|
1652 |
and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
|
|
1653 |
by blast
|
|
1654 |
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"
|
|
1655 |
proof (intro exI conjI)
|
|
1656 |
show "0<e'" .
|
|
1657 |
show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
|
|
1658 |
proof (intro strip)
|
|
1659 |
fix z::real
|
|
1660 |
assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
|
|
1661 |
with e e_lt e_simps all [rule_format, of "f x + z"]
|
|
1662 |
show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
|
|
1663 |
qed
|
|
1664 |
qed
|
|
1665 |
qed
|
|
1666 |
qed
|
|
1667 |
|
|
1668 |
theorem GMVT:
|
|
1669 |
assumes alb: "a < b"
|
|
1670 |
and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
|
|
1671 |
and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
|
|
1672 |
and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
|
|
1673 |
and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
|
|
1674 |
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)"
|
|
1675 |
proof -
|
|
1676 |
let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
|
|
1677 |
from prems have "a < b" by simp
|
|
1678 |
moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
|
|
1679 |
proof -
|
|
1680 |
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
|
|
1681 |
with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
|
|
1682 |
by (auto intro: isCont_mult)
|
|
1683 |
moreover
|
|
1684 |
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
|
|
1685 |
with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
|
|
1686 |
by (auto intro: isCont_mult)
|
|
1687 |
ultimately show ?thesis
|
|
1688 |
by (fastsimp intro: isCont_diff)
|
|
1689 |
qed
|
|
1690 |
moreover
|
|
1691 |
have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
|
|
1692 |
proof -
|
|
1693 |
have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
|
|
1694 |
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)
|
|
1695 |
moreover
|
|
1696 |
have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
|
|
1697 |
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)
|
|
1698 |
ultimately show ?thesis by (simp add: differentiable_diff)
|
|
1699 |
qed
|
|
1700 |
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)
|
|
1701 |
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" ..
|
|
1702 |
then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
|
|
1703 |
|
|
1704 |
from cdef have cint: "a < c \<and> c < b" by auto
|
|
1705 |
with gd have "g differentiable c" by simp
|
|
1706 |
hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
|
|
1707 |
then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
|
|
1708 |
|
|
1709 |
from cdef have "a < c \<and> c < b" by auto
|
|
1710 |
with fd have "f differentiable c" by simp
|
|
1711 |
hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
|
|
1712 |
then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
|
|
1713 |
|
|
1714 |
from cdef have "DERIV ?h c :> l" by auto
|
|
1715 |
moreover
|
|
1716 |
{
|
|
1717 |
from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
|
|
1718 |
apply (insert DERIV_const [where k="f b - f a"])
|
|
1719 |
apply (drule meta_spec [of _ c])
|
|
1720 |
apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g])
|
|
1721 |
by simp_all
|
|
1722 |
moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
|
|
1723 |
apply (insert DERIV_const [where k="g b - g a"])
|
|
1724 |
apply (drule meta_spec [of _ c])
|
|
1725 |
apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f])
|
|
1726 |
by simp_all
|
|
1727 |
ultimately have "DERIV ?h c :> g'c * (f b - f a) - f'c * (g b - g a)"
|
|
1728 |
by (simp add: DERIV_diff)
|
|
1729 |
}
|
|
1730 |
ultimately have leq: "l = g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
|
|
1731 |
|
|
1732 |
{
|
|
1733 |
from cdef have "?h b - ?h a = (b - a) * l" by auto
|
|
1734 |
also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
|
|
1735 |
finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
|
|
1736 |
}
|
|
1737 |
moreover
|
|
1738 |
{
|
|
1739 |
have "?h b - ?h a =
|
|
1740 |
((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
|
|
1741 |
((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
|
|
1742 |
by (simp add: mult_ac add_ac real_diff_mult_distrib)
|
|
1743 |
hence "?h b - ?h a = 0" by auto
|
|
1744 |
}
|
|
1745 |
ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
|
|
1746 |
with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
|
|
1747 |
hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
|
|
1748 |
hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
|
|
1749 |
|
|
1750 |
with g'cdef f'cdef cint show ?thesis by auto
|
|
1751 |
qed
|
|
1752 |
|
|
1753 |
end
|